Without Loss of Universality

This post is part of a series on category theory, see Overview of Blog Posts for all posts. All categories are assumed to be locally small.

Introduction

In this post, we shall establish different perspectives on the previously established concept of adjoint functors, along with more examples and properties. We shall use the same idea that underlies the Yoneda lemma to characterize an adjunction using special natural transformations called unit and counit.

To get a class of examples for adjunction, we shall study free objects in concrete categories. (Where “concrete” is a technical term. But don’t worry, the concept of concrete categories will be made concrete in examples of concrete concrete categories.).
Through them, we shall see how a natural generalization of the concept of a basis from linear algebra leads to adjunctions.

By following this approach, we will also see that there is a close connection between adjoint functors and representability.

Unit and Counit

An adjunction between two functors is given by a natural isomorphism between two functors involving Hom-functors. We know from the Yoneda lemma and in particular from its proof that a natural transformation from a Hom functor is determined by its action on identities. In the same spirit, all the information of an adjunction is contained in the action on certain identity morphisms.

Let and be categories and let be functors. Suppose we are given an adjunction, i.e. a natural isomorphism . Then for each and each , we have a bijection . If we fix and set , then we obtain an element . Similarly, we can fix and set and obtain an element .

Definition C9.1 In the above discussion, is called the unit of the adjunction (at ) and is called the counit of the adjunction.

Lemma C9.2 The collection of morphisms is natural in .

Proof Let be a morphism in , then we must show that the following diagram commutes:

i.e. .
Via naturality of in the second argument, we obtain a commutative diagram:

Applying this to the identity yields the equation , i.e. .
Naturality of in the first argument yields another commutative diagram:

Applying this to the identity leads to the equation , i.e. .
We have thus shown that , as both sides equal .

This lemma shows that the unit of an adjunction defines a natural transformation, which we denote by .

A very similar proof leads to the following lemma:

Lemma 9.3 Given the established context, the collection of morphisms is natural in .

So far, we have established how an adjunction gives rise to two natural transformations and . Following the Yoneda philosophy as described in the beginning of this section, our goal is to show that this pair of natural transformations already encapsulates the whole adjunction. To this end, we will prove properties which will turn out to give a necessary and sufficient condition for a pair of such natural transformations to come from an adjunction.

Lemma C9.4 We have for every ,

Proof Naturality of in the first argument respect to yields the commutative diagram:

Applying this to , we get , so that we obtain by the injectivity of the desired equality .

Lemma C9.5 We have for every , .

Proof Applying naturality of in the second argument with respect to yields the following commutative diagram:

Applying this to , we obtain , i.e. .

Lemma C9.6 Given a pair of functors and a pair of natural transformations , such that for all and , we have and , the map is a natural isomorphism with inverse .

Proof Naturality of follows from the naturality of . For any and , we obtain . By naturality of with respect to , we have . By assumption . Combining these, we see that . For , we have , which can be shown by analogous reasoning from the naturality of with respect to and the condition .

Putting everything together, we have shown:

Theorem C9.7 Given two categories and and functors , the datum of an adjunction between and is equivalent to giving two natural transformations , satisfying the so-called unit-counit equations: for all and , we have and .

Free Objects

We shall now return from the general to the concrete (quite literally):

Definition C9.8 A concrete category is a pair where is a category and is a faithful functor.

Using the faithful functor , which we think of as a “forgetful” functor, we can think of objects in the category as sets and of morphisms as maps. A lot of common examples of categories have a canonical choice for a faithful functor .

Example C9.9 The category of sets is concrete with the identity functor.

Examples C9.10 The categories of semigroups, monoids, groups, abelian groups, rings, commutative rings, fields, modules over a fixed base ring etc. with the respective homomorphisms as morphisms are all concrete categories where is the forgetful functor to

Examples C9.11 The category of topological spaces with continuous maps is a concrete category, so is the category of smooh manifolds with smooth maps, and so is the category of complex-analytic manifolds with holomorphic maps.

Example C9.12 Let be a monoid, considered as a one-object category. Then the idea from the proof of Cayley’s theorem from group theory allows us to realize as a concrete category: to define the functor , send the unique object of to the set underlying set of (i.e. the set of all morphisms in ) and define to be left multiplication with .

Example C9.13 Generalizing the last example, every small category admits the structure of a concrete category in a canonical way: consider the functor that sends an object to the set of all morphisms to and a morphism to the map given by post-composing with .  Then is a faithful functor, which makes a concrete category.

Example C9.14 The category of all small categories is a concrete category. We can define a functor , by sending a small category to the set of all morphisms in and sending a functor between small categories to the induced map on the sets of morphisms. (Note that the action on the objects can be recovered from the action on morphisms by looking at the images of the identities). On the other hand,the functor that sends a small category to the set of its objects is not faithful, so it does not make into a concrete category.

Remark Not every category admits the structure of a concrete category. For example, Freyd showed that the category which has as objects topological spaces and as morphisms homotopy class of morphisms is not concretizable.

Consider the notion of a basis from linear algebra. We let be the forgetful functor from the category of vector spaces over for a fixed field to the category of sets. Let be a vector space and let be a subset. To phrase the notion of a basis in terms of morphisms, one can say that is a basis for if and only for every morphism of sets to another vector space , we have a unique morphism of vector spaces such that .
This is just a formal way of saying that a subset of a vector space is a basis if and only if one can uniquely define linear maps to other vector spaces by the values of the basis vectors.

Having phrased the notion of a basis in this way, we can immediately abstract from the special case of vector spaces to arbitrary concrete categories.

Definition C9.14 Let be a concrete category. Then for an object , we say that a map is a basis for if we have the following universal property: For any object and for any map , there is a unique morphism such that . We say that an object is free if it admits a basis.

Intuitively, a basis for an object is like a subset (as we may think of objects in a concrete category as sets by the virtue of the faithful functor that is part of the datum) except that we don’t require injectivitiy, which “generates” the object in the sense that any morphism from to another object is determined by its restriction to and which does so “indepedently” or “freely” that there are no “dependencies” between the elements of in that would restrict how we can define morphisms based on where the elements in are sent.

Example C9.15 We already saw that bases in the sense of linear algebra are a special case of the abstract notion. The same holds for modules over a ring, although in general not every module is free.

Exercise C9.16 Consider the concrete category of topological spaces with the forgetful functor to , then a topological space is a free object if and only if the topology is discrete, and in this case the only possible basis is the whole space.

One can easily relate the property of a basis with representable functors:

Lemma C9.17 Let be a concrete category and let be a set. Then there exists a free object in with basis if and only if the functor is representable. In this case, a representing object corresponds to a free object and the universal element corresponds to the basis.

Proof This follows directly from lemma C5.3 in A Brief Introduction to Categories, Part 5: Universal Properties and Representable Functors.

One can observe that for modules over a ring , a module is free if and only if it is isomorphic to a direct sum of copies of and for topological spaces, a space is discrete if and only if it is homeomorphic to a disjoint union of singletons. The following lemma generalizes these obvservations.

Lemma C9.18 Let be a concrete category. Suppose that is representable by some object . Then an object is free if and only if it isomorphic to a coproduct of copies of .

Proof Let be any index set and consider the coproduct We have natural isomorphisms . We conclude by lemma C9.17 that is free with basis .
Suppose that is free with basis . Then by lemma C9.17 we have a natural isomorphism . By the computation from the other direction, the latter Hom set is naturally isomorphic to . We conclude that by the Yoneda lemma.

Example C9.19 Consider the category of small categories with the concrete category structure given by the functor that sends a small category to the set of all morphisms in . is represented by the arrow category (cf. C3.6 and C5.7). Thus lemma C9.18 implies that a small category is free if and only if it is a disjoint union of copies of the arrow category. This means that for a set , the free category generated by is given by having for each , two objects and a unique morphism (apart from the required identities).
Here’s a representation of the free category on three arrows (the identities are not depicted):

At this point, I should justify why we talk about free objects in a post about adjunctions. We will generalize the situation first.

Adjunctions via Universal Morphisms

Definition C9.20 Let be categories and let be a functor. Then we say that for an object , there exists a local left adjoint at if there is an object and a morphism with the following universal property: For every object and every morphism there is a unique morphism such that .

Local left adjoints are like bases. Clearly we have a free object in a concrete category with basis , if and only if has a local left adjoint at .

The reason for name “local left adjoint” is the following theorem:

Theorem C9.21 Let be categories and let be a functor. Then has a left adjoint if and only if for every object , has local left adjoint at .

Proof Suppose that has a left adjoint . Let be an object. Then we have the unit , so we can set , . Let be an object. As we have seen in lemma C9.6, the map is a bijection, which means precisely that for every morphism there is a unique morphism such that .
Suppose that for every object , has local left adjoint at . Call the local left adjoint of at . We need to extend the assignment to a functor. Let be a morphism in . We have morphisms and satisfying the respective universal properties. Thus we have which gives us a unique morphism such that .
We now have to check functoriality: let and suppose we are given morphisms and . By construction of and , we have and . We thus obtain . But also satisfies and so by uniqueness, we obtain . The fact that preserves identities follows by a similar argument: both and satisfy the equation .
We now have to check that is left adjoint to .
Since is a local left adjoint to for any , we have for any morphism a unique morphism such that . This implies that the map is a bijection.

In equations, for any , we need to check that
, that is
. But this follows from the functoriality of and the fact that , which was already mentioned in the construction of .

Adjunctions via Representability

Due to the close connections between representability and universal properties that were established by lemma C5.3 in A Brief Introduction to Categories, Part 5: Universal Properties and Representable Functors, we can immediately conclude from theorem C5.21:

Corollary C9.22 A functor has a local left adjoint at if and only if the functor is representable.

Example C9.23 Let be a category and let be a small category, thought of as an index category. Then the diagonal functor , which sends an object to the constant functor with value has a local left adjoint at an object if and only if the colimit over exists in .

Proof Apply the dualized version of lemma C7.10 in A Brief Introduction to Categories, Part 7: General Limits and Colimits.

Combining theorem C9.21 and corollary C9.22, we obtain:

Corollary C9.24 A functor has a left adjoint if and only if for every , the functor is representable.

Dualizing, we obtain:

Corollary C9.25 A functor has a right adjoint if and only if for each , the functor is representable.

Example C9.26 Let be a category and let be a small category, thought of as an index category. Then the diagonal functor , which sends an object to the constant functor with value has a left adjoint if and only if for all functors , the colimit exists in .  Dually, this functor has a right adjoint if and only if the limits for all exist.

Thus one can think of adjoint functors as a parametrized version of representable functors, providing yet another view on this important concept. We can also turn this around and see that representable functors are special case of adjoint functors. To see this, we use the characterization of representability via the existence of initial objects in a certain category.

Lemma C9.27 Let be a category. Then has an initial object if and only if the terminal functor has a left adjoint. Here is the category with one object and one morphism. If is a left adjoint, then evaluted at the unique object of is an initial object.

Proof Exercise. (There’s not much to check)

Corollary C9.28 A functor is representable if and only if the terminal functor has a left adjoint. Here is the category of elements, as defined here in definition C5.14. In this case, the value of a left adjoint at the unique object of is a pair , where is a representing object and is a universal element.

This follows directly from lemma C9.24 and lemma C5.15 in A Brief Introduction to Categories, Part 5: Universal Properties and Representable Functors.

This post is part of a series on category theory, see Overview of Blog Posts for a list of all posts. As always, knowing the background of every single example is not required to understand the general concept. All categories are assumed to be locally small.

Introduction

In this post, we shall study another fundamental notion of category theory that arises with remarkable ubiquity: adjoint functors. Given a functor, one can think of an adjoint as a very weak form of an inverse. There are close connections to representable functors, limits and equivalences. Knowing the basics about adjunctions of functors can be quite useful as they enjoy a non-empty set of nice properties. For illustrative purposes, we shall start by studying the special case of adjunctions between partially ordered sets, which are called Galois connections.

Galois Connections

Definition C8.1 Let and be partially ordered sets (i.e. each comes equipped with a reflexive,transitive and antisymmetric relation), then a map is called

• isotone, if for all we have .
• antitone, if for all we have .

Remark If one considers and as categories (note that every partially ordered set is in particular preordered, so cf. example C1.5), then isotone maps correspond to covariant functors and antitone maps correspond to contravariant functors.

Definition C8.2 Let and be partially ordered sets, then a covariant (or contravariant, respectively) Galois connection from to consists of the following data:

• An isotone (or antitone, respectively) map
• An isotone (or antitone, respectively) map

Such that the following condition holds:

• For all and , we have

We write

Example C8.3 Let be a map. Then the power sets and are partially ordered by inclusion, and we have isotone maps and and it holds that for all , so this is an example of a Galois connection.

Example C8.4 Let be a group and let be a normal subgroup. Then consider the partially ordered sets of all subgroups of and , respectively. Let be the canonical projection map. Then one can define maps and . As in the last example, we have , so this is an example of a Galois connection.

Example C8.5 Let and be commutative rings and let be a ring homomorphism. Then we can consider the sets of all ideals of and , respectively, denoted by and . and are partially ordered by inclusion and we have isotone maps (the ideal generated by ) , called extension and contraction. One has that for all , so this yields another example of a Galois connection.

Example C8.6 The eponymous example of a Galois connection comes from Galois theory. Let be any field extension. Then one can consider and the set of all subgroups of , denoted by and the set of all intermediate fields , both partially ordered by inclusion. Then one has antitone maps and . One checks that , so that this is a contravariant Galois connection.

Example C8.7 Now for a geometric example. Let be an algebraically closed field. Then the set of subsets is partially ordered by inclusion. On the other hand, consider the set of all ideals in , , partially ordered by inclusion. then we have antitone maps and . We have that . The sets are sets that “cut out” via polynomial equations. The example is geometric in the sense that many common geometric subsets of say (although this is not algebraically closed) can be defined via polynomial equations, such as circles, lines, parabolas, hyperbolas, conics etc.

To see that the notion of Galois connections is useful, let’s prove something.

Lemma C8.8 Let  and be partially ordered sets and , be isotone maps that form a Galois connection, i.e. . Then we have for all and for all .

Proof We have and hence as . Applying using isotonicity, we obtain . On the other hand, we get that and hence as . Thus by antisymmetry, we obtain . The other statement is proved analogously.

Corollary C8.9 Let and be partially ordered sets and , be antitone maps that form a contravariant Galois connection, i.e. . Then for all and for all .

Proof Replace by the opposite category to obtain a covariant Galois connection, now apply the previous result.

Definition C8.10 Let  and be partially ordered sets and , be either two isotone or two antitone maps that form a Galois connection. Then call closed if and only if for some . Similarly, call closed if and only if for some .

Corollary C8.11 Let  and be partially ordered sets and , be either two isotone or two antitone maps that form a Galois connection, i.e. . Then the maps and induce a bijection between the closed elements of and the closed elements of . Furthermore, is closed if and only if and is closed if and only if .

Proof Clear from lemma C8.8 and C8.9 by the equations and .

Definition C8.12 Let  and be partially ordered sets and , be either two isotone or two antitone maps that form a Galois connection, i.e. . Then the maps and are called closure operators.

Remark To motivate the terminology, note that is the smallest closed element of containing .

Example C8.13 Let be a group and be a normal subgroup and consider the Galois connection from example C8.4. As the projection is surjective, we have for any subgroup of , so every subgroup of is closed. On the other hand, it’s easy to see that for a subgroup of , we have if and only if , in general is the subgroup generated by and . It follows that and induce order-preserving bijections between subgroups of and subgroups of containing by corollary 8.11, which recovers a well-known fact from group theory.

Example C8.14 Let be a field extension, then applying corollary 8.11 to the Galois connection from example C8.6, we obtain an order-reversing bijection between intermediate extensions of the form for some subgroup and subgroups of of the form for some intermediate extension . If is algebraic and Galois, then one can show that every intermediate extension is closed, and that the closed subgroups are the subgroups which are closed in the topology which has a basis at the identity by for all intermediate extensions such that has finite degree. The correspondence between intermediate extensions and closed subgroups is the main result of Galois theory. (In the finite case, the topology is discrete and hence every subgroup is closed.)

Example C8.15 Let be an algebraically closed field and let be a natural number. Then consider the Galois connection between subsets of and ideals in . One can show that the subsets of of the form for an ideal form the closed subsets of a topology on , called the Zariski topology. Hilbert’s Nullstellensatz may be stated in the form . An ideal is called radical if . From the formalism of Galois connections, in particular corollary 8.16, we obtain an order-reversing bijection between subsets in that are cut out by polynomials and radical ideals in . The difficult part here is computing the closure operator, the rest is a formality. This is an important bridge between algebra and geometry.

Adjunctions via Hom-Sets

Definition C8.16 Let and be categories and let and be functors. Then an adjunction of the pair is given by a natural isomorphism of functors . If an adjunction exists, and are called adjoint, denoted by . In this case, is called left adjoint to and is called right adjoint to .

Exercise Verify that the notion of an adjunction specializes to that of a Galois connection in the case of partially ordered sets.

Example C8.17 Consider the category of commutative rings and the full subcategory of reduced commutative rings . We have a forgetful functor . We also have a functor that sends a ring to the associated reduced ring , the quotient by the nilradical. As every ring homomorphism from any commutative ring to a reduced commutative ring factors uniquely through , we obtain a natural bijection between ring homomorphisms and ring homomorphisms , so that the functor is left adjoint to the forgetful functor .

Example C8.18 Let and be rings and let be a -bimodule. Then we have a functor given by , this functor has a left adjoint given by the tensor product .

Example C8.19 Consider the category of small categories. We have a functor which sends a small category to its set of objects. Given a set we can form the discrete category on (see example C1.4 for details). Now a functor for some category is determined uniquely by a map : As the only morphisms in are identities, the action of a functor on morphisms is determined completely by the action on objects. Thus is left adjoint to .

Example C8.20 Let be the following category: objects are pairs , where is a commutative ring and is a multiplicatively closed subset and morphisms are ring homomorphisms such that . Let be the category of commutative rings. We have a functor sending to . The universal property of localization may be stated by saying that this functor has a left adjoint given by .

Example C8.21 Let be a complete normed field, such as , (or even , etc.). Now consider the category of normed vector spaces over with bounded linear maps as morphisms, denoted by and the full subcategory of complete normed spaces, i.e. Banach spaces over . We have a forgetful functor . We also have the completion functor . As bounded linear maps are uniformly continuous and every normed space is dense in its completion, we get that for a normed space and a complete normed space , every bounded linear map extends uniquely to a bounded linear map , where denotes the completion of . This shows that the completion functor is left adjoint to the forgetful functor .

An analogy with linear algebra

The terminology “adjunction” comes from an analogy with linear algebra. Let and be categories. Then these come equipped with their respective Hom-functors , . One can think of these Hom functor as being analogous to inner products. Then the Yoneda lemma tells us that these inner products are non-degenerate. In this analogy, for two functors and , having a natural isomorphism corresponds to having an equality for two inner product spaces and linear maps , , which is what defines to be the adjoint map to .

This post is part of a series on category theory, see Overview of Blog Posts for all posts. All categories are assumed to be locally small, unless stated otherwise.

Introduction

Previously, we have studied particular examples of limits and colimits. This time, we will go into the general theory. Using the close relation between limits and representable functors and the previous results on universal elements and representability, we are able to provide different perspectives on limits.

Limits as Universal Cones

Definition C7.1 Let be a small category, let be any category, then the diagonal functor is defined as follows:

• For any object , is the constant functor with values in , i.e. the functor which sends every object of to and every morphism in to
• For any morphism in , we define to be the natural transformation that has for any object , the component

Definition C7.2 Let be a small category, let be any category and let be a functor. Then a cone over consists of an object -called apex- and a natural transformation . The naturality condition can be stated by saying that the following diagram commutes for any morphism in :

Given two cones and , a morphism of cones is a morphism such that . Morphisms of cones can be composed and so we obtain a category of cones over , denoted by .

Definition C7.3 A limit  is a terminal object in the category of cones .

If the limit is given by a cone consisting of an object and a natural transformation , then we call the limit object (or just limit, by abuse of language) and the components of the structure morphisms.

Remark Being a terminal object, a limit is unique up to unique isomorphism of cones.

Example C7.4 Let be a discrete category, then a functor is nothing but a family of objects . In this case, a limit is just a product of the objects . As a special case, a limit over the empty functor is a terminal object.

Example C7.5 Let be a set and consider the following category , having two objects and for every , a morphism (and of course the identities for and ). There’s a unique way to define composition to turn this into a category. Then having a functor is equivalent to having a family of morphisms in with common source and target. In this case, a limit over is equivalently an equalizer for the family of morphisms .

Example C7.6 Let be the category with three objects and as morphisms (aside from the identities) a unique morphism and a unique morphism . Then a functor is equivalently a triple of objects and two morphisms and . As one might guess, a limit over such a functor is equivalently a pullback .

Example C7.7 Consider with the usual ordering as a category (this is in particular a preorder). Suppose that we have a descending sequence of sets . Then we can define a contravariant functor given by on objects and for two natural numbers with , i.e. a morphism in we let to be subset inclusion . Then the limit of the functor is the intersection , with structure morphisms being the inclusions . Indeed, for a collection of maps from an object , such that for , we have that and agree, we see that all the are equal. Hence the image is contained in and there’s a unique map that makes the necessary triangles commute. Thus we have found a terminal object in the category of cones over .

Definition C7.8 Given a functor from a small category to a category , then can also be considered as a functor , then a cocone over is defined as a cone over . A colimit is defined as an initial object in the category of cocones. Explicitly, a cocone is an object with a natural transformation . The colimit is denoted by .

Remark A limit in is a colimit in and vice versa.

Limits as Adjoints to the Diagonal Functor

Let be a small category, let be any category, then consider the following functor , denoted by .
Suppose that this functor is representable, then take a representation . Then is a limit object of and are the structure morphisms, i.e. is a terminal object in the category of cones. This is just a special case of general theorems we have proved on representable functors in virtue of the following observation:

Remark The category of cones is the opposite category of the category of elements for the functor , i.e.

Proof An object in is an object together with an element , i.e. a cone. A morphism consists of a morphism in the category of elements , i.e. a morphism in , i.e. a morphism in such that , which means that . This is precisely the notion of a morphism of cones.

Now we can apply lemma C5.15 and lemma C5.3, that is, the fact that initial objects in the category of elements for a functor and representations of that functor both correspond to universal elements to conclude the following:

Lemma C7.10 is representable if and only if the limit of in exists and representations of that functor correspond to limit objects together with structure morphisms satisfying the universal property with respect to .

General Limits from Products and Equalizers

General limits can be quite complicated. But apart from some specific examples, we don’t know yet what limits look like in familiar categories such as or if they even exist. We have already seen how pullbacks can be constructed from products and equalizers. (Cf. lemma C6.23) The following construction describes a vast generalization:

Theorem C7.11 Let be a functor from a small category to a category that has all products and binary equalizers. Then exists and can be constructed as follows: Take the product with structure maps . Now for every morphism in , we have a morphism , so that by the universal property of the product, there is a unique morphism such that for all morphisms in . Then the equalizer is the limit together with structure morphisms given by , where is the structure morphism of the equalizer.

Proof We first check that the collection of makes into a cone. Let be a morphism in . Then we have
, showing that the form the components of a natural transformation , that is we have a cone over . Let be a cone over , then for all , we have a morphism . So by the universal property of the product, we get a unique morphism such that for all . Consider the morphism . We have for any morphism in

(where we used naturality of ). So that by uniqueness of , we get that . Thus by the universal property of the equalizer , we get that a unique morphism such that .
To see that is a morphism of cones, note that . To see that is unique, let be another morphism of cones, then satisfies

Furthermore, we have , so that by uniqueness of , we get , i.e. . By the uniqueness part of the universal property of the equalizer, we get that .

This theorem not only gives us a criterion for the existence of all limits in a category from just some specific ones, but it also tells us what they look like, given that we understand products and equalizers:

Example C7.12 Consider the category of sets , then limits can be constructed as follows: be a functor from a small category , then the limit of is given by

Exercise C7.13 Using previously estabished contructions of coproducts and coequalizers in , describe general colimits in .

Limits as Representations of the Limit Functor

We have already seen that for a functor from a small category to any category , the existence of a limit is equivalent to the representability of the functor . In this section we shall derive another description of this functor that relates limits in arbitrary categories to limits in .

Definition C7.14 Let be a small category, be any category and be a functor. Then the functor is defined as follows: for an object , we can consider the functor , (i.e. we compose with the covariant Hom-functor corresponding to ). Since limits in exist, we can take the limit . This is the value of the functor at . (Exercise: Check that this is indeed a functor, i.e. that it acts in a compatible way on morphisms)

Remark This is actually the limit in the functor category. One can check that limits in functor categories may be computed “pointwise”.

Lemma C7.15 There’s a natural isomorphism of functors : Both functors send an object to the set of cones with as an apex.

Proof By definition, an element of is a cone with is an apex. For , we can use the description of example C7.12:  
Clearly this set consists of all natural transforations . Naturality is easily checked: Indeed, using the now established identifications on objects with the functor sending an object to the set of cones with as an apex, both functors send a morphism to the map from the set of cones over with as an apex to the set of cones over with as an apex that takes a cone to the cone with as an apex with the natural transformation whose components are for each .

Corollary C7.16 A limit of a functor from a small category to exists if and only if the limit functor is representable.

Corollary C7.17 Covariant Hom-functors are “continuous” in the following sense: if a limit over a functor exists, where is a small category and is any category, then

Corollary C7.18 Dualizing the previous corollary, contravariant Hom-functor send colimits to limits.

This post is part of a series on category theory. See Overview of Blog Posts for a list of all posts. All categories in this post are assumed to be locally small (the morphisms between two objects form a set), unless stated otherwise.

Introduction

Last time, we studied universal properties and representable functors. A particular class of universal properties and representable functors is given by (co)limits. This class generalizes many known and frequently appearing constructions and is of great importance both for the study of categories themselves and for applications of category theory. Using the notion of limits, we will be able to state and prove a useful criterion for when a functor is representable in a future post. For motivation, we will start be studying particular examples of limits and colimits in familiar categories before giving the general definition in the future. As we shall see in future posts, all general limits and colimits can be constructed from these examples.

Terminal and Initial Objects

We have already defined initial objects in definition C5.12, but for completeness’s sake, we shall repeat the definition here.

Definition C6.1 Let be a category. Then an object is called terminal if for any object , there is a unique morphism .

Definition C6.2 Let be a category. Then an object is called initial if for any object , there is a unique morphism .

Remark From the definitions, it is clear that an initial object is just a terminal object in the opposite category and vice versa.

Remark By lemma C5.13, intial objects (and hence terminal objects, see above) are unique up to unique isomorphisms.

Example C6.3 In the category of sets, the empty set is the unique initial object. Indeed, for any set , there is a unique map , namely the empty map. Every one-point set is a terminal object, as for any set , there is a unique map that sends everything to . Similarly, in the category of topological spaces, the empty space is an initial object and the one-point space is a terminal object. In the category of small categories , the empty category is an initial object and the category with one object and one morphism (the identity of the one object), i.e. a one-object discrete category, is the terminal object.

Example C6.4 In the category of groups, abelian groups, or modules over a ring , the trivial group (or zero module module) is both an initial and a terminal object.

Example C6.5 In the category of rings, the zero ring is a terminal object and the ring of integers is an initial object. More generally, let be a commutative ring and consider the category of -algebras. Then is an initial object and the zero ring is a terminal object. This generalizes the last example, as rings are equivalently -algebras.

Example C6.6 Let be a preordered set, considered as a category (cf. example C1.5). Then an initial object is an element , such that for all , i.e. a least element. Dually, a terminal object is a greatest element.

Example C6.7 The category of fields does not have an initial object or a terminal object. The reason is that if two fields have different characteristic, there is no field homomorphism between them. If we fix the characteristic to be or a prime number , then the category of fields of that characteristic has the prime field of that characteristic, i.e. or as an initial object, but it still has no terminal object, as all field homomorphisms are injective and there are fields of arbitrarily large cardinality.

Remark In the terminology of universal properties and representable functors, an initial object in represents the constant functor that sends every object to the terminal object in , namely a one-point set. The identity of the initial object is a universal element for this constant functor. Dually, a terminal object represents the constant functor that sends everything to a one-point set, considered as a contravariant functor, i.e. a functor . The identity of the terminal object is a universal element.

Products and Coproducts

Definition C6.8 Let be a set and let be a collection of objects in a category . Then a product is given by the following data:

• An object in , denoted by
• A collection of morphisms, for any , a morphism , called structure morphisms or canonical projections.

With the following universal property:

• For any object in and a collection of morphisms , there exists a unique morphism such that for all .

The property can be visualized via a commutative diagram, here shown in the case of binary products:

Remark C6.9 Let be the empty set. Then a product is exactly a terminal object.

Example C6.10 Consider the category of sets . For a family of sets , one can construct the Cartesian product and let be the projection onto the -th coordinate. This clearly satisfies the universal property for a product, as for any set and maps , the obvious definition yields the unique map such that for all . If we have the category of semigroups, of monoids,  of groups, of abelian groups, of rings, of -modules or of -algebras etc. with their respective homomorphisms as morphisms, then equipping the cartesian product with coordinate-wise operations yields the product in that category. (Note that as the operation in the product is defined coordinate-wise, the projections are homomorphisms of the respective type of algebraic structure.)
In the category of topological spaces , one takes the Cartesian product and equips it with the product topology to obtain the categorical product.

Example C6.11 Let be a preordered set. Then the product of a family of objects, if if it exists, is the infimum. The universal property of the product turns into the universal property of the infimum: . For example, if is an integral domain and we define if and only if divides , then the product will be the greatest common divisor of a family of objects, if it exists.

Example C6.12 Consider the category of small categories . Then for a family of categories , there’s the product category together with projection functors for all . The objects are given by and the morphisms are given by with composition defined component-wise. We have already seen the special case for binary products in definition C3.7.

Remark Let be a category and let be a family of objects indexed by a set . Then if the product , exists, we have a bijection for any object ,, given by , where the are the structure morphisms of the product. One checks that this is natural in . This means that the functor is represented by and that the family of structure morphisms is a universal element. Thus we have shown that the “universal property” of the product is indeed a universal property in the formal sense defined in the last entry.
Conversely, given a representation of the functor , the representing object is a product of the and the universal element is the family of structure morphisms.
Being representing object of a functor, a product is uniquely determined up to unique isomorphism in the following sense: If and are products for the family with structure morphisms and , then there exists a unique isomorphism such that for all .

As with any notion defined for a category, we can dualize the concept of a product.

Definition C6.13 Let be a category, let be a set and let be a family of objects in . Then a coproduct of is a product of in . The coproduct is denoted by , the structure morphisms are also called canonical inclusions.

Exercise Write out the universal property of coproducts as in definition C6.8.

Remark Dualizing a previous remark, a coproduct over an empty set is equivalently an inital object.

Example C6.14 Consider the category of sets . The coproduct of a familiy of sets is given by the disjoint union and the structure morphisms are given by the inclusions . The universal property of the coproduct says that having a map defined on a disjoint union is equivalent to having a map for each . The correspondence is given by composing with inclusion, i.e. restricting to each component of the disjoint union.
In the category of topological spaces, we can also take the disjoint union with the finest topology that makes all inclusions continuous.
Similarly, in the category of small categories, one can take the disjoint union of objects and the disjoint union of morphisms to get coproducts. (E.g. any discrete category is a disjoint union of copies of the terminal category. The category of fields is a disjoint union of the categories of fields of a fixed characteristic , where varies over all primes and .)

Example C6.15 Let be a ring and consider the category of left -modules . For a family of left -modules , the direct sum (i.e. the submodule of the product consisting of all tuples in which all but finitely many entries are ) is the coproduct with the structure morphisms given by the inclusion in the j-th coordinate. Note that for a finite family, product and coproduct agree, although the structure morphisms are different.

Example C6.16 Consider the categeory of groups . Here the coproduct is given by the so-called free product. For simplicity, let us define the free product in terms of presentations (as it satisfies a universal property, it is unique up to unique isomorphism and hence doesn’t depend on the presentation.) If is a family of groups with presentations , then the free product has a presentation and the structure morphisms are induced by the inclusion of generators . (cf. example C5.5 for the universal property of presentations.)

Example C6.17 Dualizing example C6.11, if we consider a preordered set as a category, then the coproduct of a family of objects is given by the supremum, if it exists. For the divisibility relation in an integral domain, this is the least common multiple.

Remark Just like with products, coproducts can be understood in terms of representable functors (they are just products in the opposite category, after all). Explicitly, for a family of objects in a category , we can form the product of Hom-functors . A representation of this functor is equivalently a coproduct, with the universal element being the family of structure morphisms. (cf. the corresponding remark for products.)

Equalizers and Coequalizers

Definition C6.17 Let be a category, let be objects and let be a family of morphisms. Then an equalizer consists of the following data:

• An object in
• A morphism , called structure morphism or -by abuse of terminology- also the equalizer of the , which satisfies .

Such that the following universal property holds:

• For any object and every morphism such that , there is a unique morphism such that

The universal property can be visualized via a commutative diagram, here shown in the case of binary equalizers:

Remark Equalizers are terminal objects in a suitable chosen category: in the situation of definition C6.17, define the following category :

• Objects are pairs where is an object in and is a morphism such that .
• Morphisms are morphisms in such that
• Composition is inherited from

Then one may restate the definition of an equalizer by saying that an object is a terminal object if and only if is an equalizer with structure morphism . This implies that an equalizer is unique up to unique isomorphism in the following sense: If and are both equalizers of the same family of morphisms , then there exists a unique isomorphism such that .

Example C6.18 Consider the category of sets . Then an equalizer of a family of maps can be realized as the largest subset of on which they agree: If we let be the subset of consisting of all such that  and take to be the subset inclusion, then any map  such that has its image contained in . Hence it factorizes uniquely through the subset inclusion , showing that satisfies the universal property. If all sets involved are semigroups, monoids, groups, rings or -modules etc. and the maps are homomorphisms of the respective type, then will be a subsemigroup, submonoid, subgroup, subring or submodule, respectively and by the same argument, satisfies the universal property of the equalizer.
Suppose a group acts on a set , then we can consider the action as a collection of maps . In this case, one checks that the set is given by the set of invariants .

One can think the equalizer as measuring “how equal” a collection of morphisms are.

Definition C6.19 A coequalizer in a category is an equalizer in the opposite category .

Exercise Write out the universal property for coequalizers.

Example C6.20 Consider the category of sets , then a coequalizer of a family of morphisms , can be constructed as follows: Consider the equivalence relation on generated by for all and . (Given a relation on a set, the intersection of all equivalence relations containing it is again an equivalence relation, called the equivalence relation generated by it.) Then the quotient map is a coequalizer. If a group acts on a set , then if we consider the coequalizer of the family of maps by which acts, this is the quotient space . If we use the quotient topology, the same construction works in the category of topological spaces.

Example C6.21 If we work in the category of -modules and we have a family of -linear maps , then one can take the quotient of by the submodule generated by the set where vary over and varies over . The quotient module with the quotient map is then a coequalizer in the category of -modules.

If one thinks of an equalizer as a universal solution to the problem of making maps equal from the left (i.e. by precomposing with a map), then dually a coequalizer is a universal solution to the problem of making maps equal from the right (i.e. by postcomposing with a map).

Pullbacks and Pushouts

Definition C6.22 Let be a category, let be objects and let and be morphisms. Then a pullback consists of the following data:

• An object
• Morphisms

Such that the following universal property holds:

• 
• For every object and morphisms and such that , there exists a unique morphism such that and

The universal property can be visualized by the following commutative diagram:

Exercise Generalize this definition to the case of a family of objects and a family of morphisms .

One can think of pullbacks as a “mix” of products and equalizers. For products, we had multiple objects but no morphisms between them as a given datum, whereas for equalizers, we had multiple morphisms, but all with the same source and target. For pullbacks, we have different objects and morphisms from those objects to a common target. The following lemmata make the relation of pullbacks to equalizers and products precise.

Lemma C6.23 In the situation of definition C6.22, suppose that the product exists and suppose that for the structure morphisms of the product and , the equalizer exists with structure morphism . Then this equalizer is a pullback with structure morphisms and .

Proof We verify the universal property. We have from the definition of an equalizer. Let be an object and let and be morphisms such that . Then by the universal property of the product, there is a unique morphism such that and . Then we get that , so that by the universal property of the equalizer, there is a unique map with the required property.

Lemma C6.24 Suppose that a category has a terminal object . Then, if for two objects , the pullback (along the unique morphisms ), exists, it is a product of and .

Lemma C6.25 Suppose that in a category , the product for an object exists. Let be two morphisms from another object and let be the unique morphism such that and . Furthermore, let be the unique morphism such that and . Then if the pullback exists, it is an equalizer with structure morphism defined via the projection .

Exercise Prove the preceeding two lemmata. (This is a nice exercise in getting used to working with universal properties.)

Exercise If a pullback exists, it is unique up to unique isomorphism in a certain sense. Specify what that means and prove it. (There are a couple of different ways to see this, one can define a category such that a pullback is a terminal object (as we did for equalizers), one can define a functor such that a pullback is a representation of that functor (as we did for products), or one can work with the universal property directly.)

Example C6.26 Lemma C6.23 tells us what pullbacks look like, given that we know products and equalizers. For example, in the category of sets, a pullback of two maps and can be realized as a subset of the product . The same construction works for semigroups, monoids, groups, rings etc.

As you might expect, we can dualize this notion, giving rise to the notion of pushouts.

Definition C6.27 A pushout in a category is a pullback in the opposite category

Example C6.28 Consider the category of commutative rings. Given commutative rings and ring homomorphisms , , we can consider both and as -modules via and , respectively, so we can consider the tensor product . Via the definition on elementary tensors, one obtains a multiplication (using the universal property of tensor products to see that it is well-defined) that turns into a commutative ring. The structure morphisms are given by and . Using the dualized version of lemma C6.24, we see that the coproduct of two commutative rings and is given by .

This concludes this post on examples of limits and colimits, next time we will look into the general definition.

This post is part of a series on category theory. See Overview of Blog Posts for all posts.

Introduction

Many objects or contstructions can be characterized by a so-called universal property that describes either the morphisms to them or the morphisms from them. By the virtue of the Yoneda lemma, we can forget about any concrete representation of the object and can only work with its universal property. One can think of a universal property as a “universal” solution to a problem in the sense that it is uniquely related to all other solutions. For example, consider the problem of finding a commutative ring and two elements such that . The universal solution to this problem is given by the elements and in ; this solution is universal in the sense that for every other commutative ring and elements such that , there’s a unique ring homomorphism such that and . The general definition is a straightforward abstraction from this example. Universal properties are closely related to representable functors and this relation is established by the Yoneda lemma. Understanding objects in terms of universal properties is the epitome of categorial thinking.

Representability and Universal Elements

Throughout this section, let be a locally small category and let be a functor.

Definition C5.1  is called representable if there is natural isomorphism for some object . We say that is represented by and call a representing object. The pair is called a representation of .

Definition C5.2 For an object , we say that an element is universal if the following property holds: For any object and , there’s a unique morphism with . In this case, this property is called the universal property of .

Lemma C5.3 For an object , the representations are in bijection to universal elements of the form via

Proof Via the Yoneda lemma, we know that natural transformations correspond to elements in via . We need to check that is an isomorphism if and only if is an universal element. For every object , we have , but by naturality of , we have for any morphism , (cf. the proof of the Yoneda lemma). Thus is a natural isomorphism if and only if for every object , the map is a bijection, which is just a restatement of saying that is a universal element.

Example C5.4 Let be a commutative ring and consider the forgetful functor from the category of commutative -algebras to the category of sets , that sends an -algebra to its underlying set. Then is the represented by the polynomial ring . Concretely, we have a natural isomorphism . is the corresponding universal element. This holds, because any -algebra homomorphism is given by evaluation at a uniquely determined element, namely .

Example C5.5 Let be a group with a presentation with being the set of generators and the set of relations. Then for if we have for a group for every generator an element such that the relations in are satisfied by those corresponding elements, this uniquely determines a group homomorphism . Consequently, for the functor sending a group to the families of elements in indexed by such that the corresponding relations are satisfied, we see that the family in is an universal element and is represented by .

Example C5.6 Let be a ring and , then consider the functor , i.e. from the category of left -modules with -linear maps to the category of sets that sends a module to the set . Then this functor is represented by and is a universal element.

Example C5.7 Let be the functor from the category of small categories to the category of sets that sends a small category   to the set of all morphisms in . Then is represented by the arrow category from definition C3.6. The unique morphism in the arrow category is a universal element.

Example C5.8 Let be a small category, let be an object and consider the functor from the functor category to sending a functor to and a natural transformation to . Then the Yoneda lemma in this context can be stated by saying that this functor is representable by and the identity is a universal element.

Example C5.9 Let be a group, not necessarily abelian. Then consider the functor from the category of all abelian groups to the category of sets given by taking all group homomorphism to an abelian group: (this is a hom-functor from a larger category restricted to a subcategory). Then this functor is representable by the abelianization with the quotient map as a universal element in : This is the universal property of the abelianization: Every map from to an abelian group factorizes over the quotient map .

Example C5.10 To give an example of a contravariant representable functor, consider the functor from the opposite category of topological spaces to the category of sets that sends a topological space to the set consisting of all of its open subsets. A continuous map is sent to the map . This functor is represented by the Sierpinski space whose points are with the topology that has as open sets . (For all algebraic geometers: this arises as the spectrum of a discrete valuation ring.) The open subset is a universal element. This reason this holds is that for any map , we always have and , so the only interesting part of being continuous is requiring that is open. Since has only two points, knowing completely determines . Putting these things together, we get that the map is a bijection, proving the claim.

Example C5.11 Not every functor is representable. As a boring example, one can take the constant functor that sends every object to the empty set. (Exercise: why will this never be representable, provided that is non-empty?) For a more interesting example, consider the functor from the category of all groups to the category of sets that sends a group to the set of all of its torsion elements, i.e. elements of finite order. . (This defines a functor because group homomorphisms send torsion elements to torsion elements.) Suppose that is represented by a group and let be a universal element. Then for every , we have an element . By universality of , there is a group homomorphism that sends to . But this implies that divides the order of . As was arbitrary, this is impossible, as the order of is finite.

Universal Elements as Initial Objects

One can ask for a characterization for representability of a functor that doesn’t refer to a representing . In this section, we shall introduce the notion of an initial object to obtain such a criterion, although the criterion will be a mere reformulation of the definition. This reformulation will be put to use in the future to prove a more useful criterion, for which we lack the tools at the moment. We shall return to the notion of initial objects later when we generalize them vastly to colimits.

Definition C5.12 Let be a one-point set. An object in a category is called initial if it represents the constant functor , given by sending each object in to and every morphism to .

Unfolding the definition, this means that for an object is initial if for any object , there is exactly one morphism

We prove an elementary, but useful property of initial objects:

Lemma C5.13 If a category has an initial object, it is unique up to unique isomorphism.

Proof Let and be initial objects in , then there are unique morphisms and . Then both and are morphisms , so they must agree, thus . By symmetry, proving that and are mutually inverse isomorphisms. is necessarily unique as is initial.

Definition C5.14 Let be a functor where is a locally small category. Define the category of elements as follows:

• Objects are pairs consisting of an element and an element .
• Morphisms are morphisms in such that that .

Lemma C5.15  satisfies the universal property for if and only if is an initial object in .

Proof Immediate from the definitions.

Corollary C5.16  Object characterized by a universal property are unique up to unique isomorphism in the following sense: if and both satisfy the universal property for a functor , then there exists a unique isomorphism such that .

Proof By lemma C5.15, both and are initial objects in . Now apply lemma C5.13.

 

This concludes this post on representability and universal properties. Next time, we will see an important class of examples for those concepts, given by (co)limits.

This post is part of a series on category theory, see Overview of Blog Posts for all blog posts.

Introduction

Using the notion of natural transformations introduced previously, we are able to state and prove the (in)famous and all-important Yoneda lemma, which is not only applied throughout category theory, but can yield satisfying conceptual proofs in many distinct areas. This result single-handedly justifies the “behavioristic” or “relative” viewpoint internal to category theory that lies in not bothering with concrete qualities of the objects one is interested in and studying them via their relations to other objects, for it implies in a certain technical sense that an object is uniquely determined by how it relates to other objects. This allows us to think of objects in terms of functors.

Functor Categories

Let and be categories, then we can consider the class of functors . For two such functors, we can consider natural transformations between them, which we can compose in an associative way. For any functor, there’s the identity natural transformation. So this turns the collection of functors into a category, right?
Not quite, at least with our definitions. The subtle problem is that there is no reason to expect that the natural transformations between two functors form a set, there might be “too many”. To remedy this deficiency, we shall amend the definition of a category. Henceforth, what is defined in definition C1.1 shall be called a “locally small category” and a not necessarily locally small or “large” category has the same definition, except that the morphisms between two objects are allowed to be proper classes. With this amendment, we can make the following definition:

Definition C4.1 Let and be categories. Then the functor category, denoted by has as its objects all functors and as morphisms natural transformations, with composition of natural transformations defined pointwise.

Exercise C4.2 If is a small category (the objects form a set and it is locally small) and is a locally small category, then the functor category is locally small.

Example C4.3 As we have already seen, the functor category for a group considered as a one-object category is the category of -sets with equivariant maps as morphisms. (cf. example C2.4 and example C3.3)

Example C4.4 Similarly, for a field and a group , the functor category is the category of representations of over . (cf. example C2.5 and example C3.4)

The Yoneda Lemma

We now come to the promised infamous lemma:

Theorem C4.5 (Yoneda lemma) Let be a locally small category, let be an object in and let be a functor, then the map is a bijection that is natural in and in .

Proof construct an inverse map as follows: let , then set to be the following natural transformation: for any object and any morphism , let , we need to check that this is indeed a natural transformation from to . Let be a third object and let be a morphism in . Then we need to verify that the following diagram commutes:

To show this, let , then showing that the diagram commutes. Hence is a natural transformation.

We now need to show that is the inverse of . Let be a natural transformation, then we need to show that , which we can check this component-wise. Let be an object and let be a morphism, then we have by naturality of this equals , see the diagram below, applied to the identity on :

This proves one direction, for the other direction, we have for , . This shows that and are mutually inverse bijections. Naturality in and are easy computations.

There is also a contravariant version of the Yoneda lemma:

Corollary C4.6 Let be a locally small category and let be a functor, i.e. a contravariant functor on , then for any object , the map is a natural bijection.

Proof Apply the Yoneda lemma to the opposite category.

The most important case is when is another Hom-functor:

Corollary C4.7 Let be a locally small category and let be objects. Then there is a natural isomorphism given by .

Corollary C4.8 Let be a locally small category. Then the functor , called the Yoneda embedding, given by is fully faithful.

Corollary C4.9 Let be a locally small category, then if for two objects and , the functors and are naturally isomorphic, then and are isomorphic up to a unique isomorphism inducing a given natural isomorphism between the functors.

Proof By corollary C4.8, the Yoneda embedding is fully faithful, hence it reflects being isomorphic by lemma C3.11. As the Yoneda embedding is faithful, we get uniqueness.

Corollary C4.10 Let be a locally small category. If for two objects and , the functors and are naturally isomorphic, then and are isomorphic up to a unique isomorphism inducing a given natural isomorphism between the functors.

Proof Apply corollary C4.9 to the opposite category and note that two objects are isomorphic if and ony if they are isomorphic in the opposite category.

Remark Using the Yoneda lemma, one can identify a locally small category with the subcategory of the functor category , containg as objects all Hom-functors and as morphisms natural transformations between them.

Let us ponder for a moment what a remarkable thing we have shown: The last two corollaries tell us that in order to study an object, it is sufficient to study all morphisms from that object or all morphisms into that object. To put it succinctly, the Yoneda lemma tells us that an object is determined by its relation to other objects. Studying the functor or the functor can be easier than studying an object itself, especially if the objects satisfy so-called universal properties, which are basically convenient descriptions of the functors from or to an object.

Example C4.11 Let be the category of commutative rings, then if is a commutative ring and is an ideal, one has an isomorphism natural in : , given by pulling back along the projection . In this sense, the projection is the universal morphism that restricts to on : every other such morphism factorizes over it. This is the content of the homomorphism theorem. From a categorical perspective, the homomorphism theorem is the proper definition of a quotient modulo an ideal, whereas the construction with equivalence classes is mere coincidence. In a similar manner, for a multiplicatively closed subset , the localization together with the canonical morphism is the universal solution to the problem of finding a ring and a ring homomorphism such that maps all elements of to units. Explicitly, one has a natural isomorphism .
If we look into the case that we have both an ideal and a multiplicatively closed subset in , then we can ask for a universal morphism that sends both to and maps to units. Here we can see the Yoneda lemma in action, for we have natural isomorphisms
which implies that by the Yoneda lemma. This is a conceptually very satisfying proof, as we didn’t have to fiddle with equivalence classes or fractions, but we only used the abstract defining properties of localizations and quotients.

Exercise Use the Yoneda lemma to prove that if is a commutative ring and and are two ideals such that , then .

Exercise If and are two multiplicatively closed subsets of a commutative ring and is the multiplicatively closed subset generated by and , then show that using the Yoneda lemma.

This concludes the post on the Yoneda lemma, which we shall see in many applications to come.

This post is a continuation of a mini-series on category theory, starting with this post. As before, knowing the background for every single example is not required for understanding the overall concepts.

Introduction

Previously, we have defined categories and the morphisms between them. In the case that one has two parallel functors , between two categories and , it is frequently the case that for any object , one has a morphism . In this situation, the notion of a natural transformation conceptualizes what it means for this collection of morphisms to vary “naturally” with the object , leading to a concept essential to all of category theory. Using this concept, we can define a coarser notion of equivalences than the obvious one of isomorphic categories.

Natural Transformations

Definition C3.1 Let and be categories and let be functors. Then a natural transformation consists of the following data:

• For every object a morphism in , called the component of at .

Such that the following condition holds:

• For every objects and every morphism , the following diagram commutes:
   which means that

One can think of the naturality condition as saying that “interpolates” between and .

Definition C3.2 If all the components of a natural transformation are isomorphisms, is called a natural isomorphism and and are called naturally isomorphic.

Example C3.3 Let be a group and and be -sets, considered as functors . Then a natural transformation has only one component, as has only one object, so it is given by a map of the underlying sets satisfying a naturality condition. This condition requires that we have for any (i.e. for any morphism in ) , (by abuse of notation, we write for its action on and ), so that natural transformations correspond precisely to -equivariant maps.

Example C3.4 Reasoning as in the last example, one concludes that if we have a group and a field , then natural transformations between two representations, considered as functors correspond to morphisms of representations.

Example C3.5 Let be the category of commutative and unital rings with ring homomorphisms and be the category of monoids. Fix . Let be the functor that sends a commutative ring to the multiplicative monoid of the matrix ring: . Every ring homomorphism induces a monoid homomorphism of multiplicative monoids by applying to every component (this is in fact a ring homomorphism of the matrix rings). Let be the “forgetful” functor that forgets about addition and sends a commutative ring to its multiplicative monoid. Every ring homomorphism respects multiplication and the unit element, so that it induces a monoid homomorphism on the multiplicative monoids. The determinant is defined by a polynomial with coefficients in . Hence it doesn’t matter if we apply the ring homomorphism componentwise to a square matrix and then take the determinant or if we take the determinant first and then apply to the result. We hence get a commutative diagram:  stating that the determinant is natural in .

Natural Transformations as Homotopies

(Image source: https://commons.wikimedia.org/wiki/File:HomotopySmall.gif)

Reminder/Definition Consider two topological spaces and and let and be continuous maps, then a homotopy between and is a continuous map such that for all and for all . If there exists such a homotopy, and are called homotopic, denoted by .

One thinks of a homotopy as a continuous deformation or an interpolation of to and of the first parameter from as a time parameter, as in the picture above. The definition states that behaves on like and on like . Continuity implies that it is a continuous deformation of to .

If we think of categories as the objects of study in category theory and functors as the morphisms between them, then natural transformations interpolate between  morphisms between the objects. This situation might be reminiscent of the situation in topology: objects in topology are topological spaces, the morphisms between them are continuous functions and homotopies interpolate between continuous functions. We will see in this section that this analogy can be made even closer than this vague similarity by defining the analogue of products and the unit interval for categories.

Definition C3.6 The arrow category or interval category is the following category: we have two objects and three morphisms: the two identities and a unique morphism . One easily observes that there is only one possible way to define composition which makes this into a category.

This category plays a role for natural transformations analogous to the unit interval

Definition C3.7 Let and be categories, then the product category consists of the following data:

• 
• For and , we have
• Composition is defined componentwise.

Using these definitions, we can define the analogue of a homotopy: let and be categories and let and be functors. Then a “categorical homotopy” from to is a functor such that for all and for all and for all morphisms in , we have and . This means that on the subcategory , behaves like and on the subcategory , behaves like .

Proposition C3.8 Let be categories and let be functors, then natural transformations correspond to categorical homotopies from to .

Proof Let be a natural transformation, then we define a categorical homotopy as follows:

• For all , let for and .
• For all morphisms in , let and let and let (the last equality holds by naturality of .)

Let us check that this is indeed a functor. It clearly respects identities. The only non-obvious cases for compatibility with composition involve in the first component. We have for morphisms and in ,

In the other case we have

Such that is indeed a functor. It is clear from the construction that is a categorical homotopy from to , completing one half of the correspondence. For the other direction, let be a categorial homotopy from to . To obtain a natural transformation, set for all . Naturality now follows from functoriality of ,
as for any morphism in , we have:

Using the same computations, one checks that those two constructions are inverse to each other.

Equivalences of Categories and Related Properties

There’s an obvious notion of isomorphisms for categories: a functor is an isomorphism if there is a functor such that and are the identity functors on and , respectively. For many purposes, however, this notion is too strict and a coarser notion is much more useful. We can let the analogy between natural transformations and homotopies guide us.

Reminder/Definition A continuous map between topological spaces and is called a homotopy equivalence if there is a continuous map such that and are homotopic to the identities on and , respectively.

Example The -dimensional sphere is homotopy equivalent to .

This leads “naturally” to the following definition:

Definition C3.9 A functor between two categories and is called an equivalence of categories if there is a functor such that and are naturally isomorphic to the identity functor on and on , respectively.

It is sometimes easier to verify some sufficient (and necessary) conditions for a functor to be an equivalence than to use the definition directly by constructing a functor in the other direction. We shall now define those properties, which are important in their own right.

Definitions C3.10 A functor between two categories and is called

• faithful, if for all , the induced map is injective.
• full, if for all , the induced map is surjective.
• fully faithful, if for all , the induced map is bijective.

We state and prove a useful property of fully faithful functors:

Lemma C3.11 Let be a fully faithful functor, then reflects isomorphisms in the following sense: if are two objects such that and are isomorphic in , then and are isomorphic in .

Proof Take an isomorphism with inverse , then there are some such that and as the induced maps on -sets is surjective. We get , so as is faithful. By symmetry, , so is an isomorphism with inverse .

Lemma C3.12 A natural isomorphism of functors preserves fullness, faithfulness and full faithfulness

Proof Let be functors and let be a natural isomorphism. One immediately sees that for any objects the map is a bijection with inverse given by .
By naturality, we have for any morphism in , which implies that From this equation, we conclude that the maps induced by and , respectively, on Hom-sets are related by the bijection , hence one is injective, surjective or bijective if and only if the other one is.

Lemma C3.13 If and are functors such that is faithful, then is faithful. If is full, then is full. If is fully faithful, then is full/faithful/fully faithful if and only is.

Proof These assertions follow immediately from the corresponding elementary statements about injective, surjectiv and bijective maps. (E.g. if is surjective, then is surjective.)

Corollary C3.14 Any equivalence of categories is fully faithful.

Proof Let be a functor and be a functor such that is naturally isomorphic to the identity functor on and is naturally isomorphic to the identity functor on . By using lemma C3.12, we can conclude that and are fully faithful, as the identity functor is evidently fully faithful. In virtue of lemma C3.13, this implies that is both full and faithful, hence fully faithful.

Definition C3.15 A functor is called essentially surjective if for every object , there is an object such that is isomorphic to .

Lemma C3.16 An equivalence of categories is essentially surjective.

Proof Let be a functor and be a  functor such that is naturally isomorphic via to the identity functor on . Then for any object , we have an isomorphism , showing that is essentially surjective.

At this point, we have collected enough necessary properties of equivalences of categories to obtain a characterization that doesn’t rely on the existence of another functor, which consitutes the main result of this post.

Theorem C3.17 A functor is an equivalence of categories if and only if it is fully faithful and essentially surjective.

Proof Lemma C3.16 and corollary C3.14 furnish one half of the proof.
For the other half, let be a fully faithful and essentially surjective functor. For any object , choose an object such that is isomorphic to (which is possible, as is essentially surjective.) Choose an isomorphism . Let be a morphism in . Then is a morphism . By full faithfulness of , there is a unique morphism such that .
We need to check that these assignments make into a functor: let and be morphisms in , then from the definition of , we obtain , such that by faithfulness of , . As for identities, we get for any object , so that by faithfulness, .
We now need to show that and are naturally isomorphic to the respective identities: The defining equation yields so that is a natural isomorphism from the identity functor on to .
.
To finish the proof, we have to show that is naturally isomorphic the identity functor on . Note that since is naturally isomorphic to the identity functor on we get that is naturally isomorphic to , because we have for any object an isomorphism such that for any morphism in , we have . Now as is fully faithful, there exist for each pair of objects a unique morphism such that . To show that is natural, insert into so that the definition of , so that we get which implies by functionariality so that by faithfulness, we get , which means precisely that is a natural transformation from the identity functor to . As is an isomorphism for each and is fully faithful, one concludes that is an isomorphism. (If is a fully faithful functor and is an isomorphism, then is an isomorphism, cf. the proof of C3.11.) This shows that is naturally isomorphic to the identity functor on , which concludes the proof and also this post.

(Exercises and remarks for the stray logician or set theorist:
The above proof doesn’t work in ZFC, why? Give an example of an underlying set theory as a meta-theory such that the proof does indeed work.
Prove that the above proof does work in ZFC if one restricts the statement to small categories and that the statement for small categories is equivalent to choice over ZF.
In practice, set-theoretic issues are often ignored by those doing category theory, or one uses Tarski-Grothendieck set theory)

This post is a continuation of a previous post and part of a mini-series on category theory. As before, knowing the background of every single example is not necessary (nor sufficient) to grasp the general concept.

Introduction

Previously, we have defined and discussed categories. However, studying categories as isolated objects with no relation to other categories would be a fruitless toil. It would also not be in the spirit of category theory, which studies everything through its relationship to other things. Yet the necessity to bring together distinct instances of the objects one is studying is not specific to category theory: For example, it’s hard to imagine studying group theory without studying group homomorphisms.

It should by now be clear that we are in dire need of a notion allowing us to connect different categories and to express relationships between them. It seems natural to let the definition of a category guide us to the definition of things that relate them. This plan comes to fruition in the conception of functors. These should be thought of as (homo)morphisms of categories, mappings preserving the structure categories have, i.e. identities and composition, just like a monoid homomorphism preserves the neutral element and the monoid operation.

The ubiquity with which functors arise is remarkable, and after some experience in thinking categorically it will become second nature to ask whether a construction behaves “functorially”.

Functors

Definition C2.1 Let and be categories, then a (covariant) functor consists of the following data:

• A map
• For any pair of objects a map

such that the following conditions hold:

• For every object we have
• For any three objects and morphisms we have .

 

Let us ponder for a moment how this definition generalizes familiar concepts.

Example C2.2 Let and be sets, then we can consider them as discrete categories (cf. C1.4), so that the elements of each set are the objects of the associated discrete category and the only morphisms are the identity morphisms. Then any map uniquely determines a functor between the associated discrete categories restricting to on the objects, as the only morphisms are identities and the definition of a functor requires sending identity morphisms to the corresponding identity morphisms in the target category. To summarize, functors between discrete categories correspond to maps between the underlying sets.

Example C2.3 Let and be monoids, considered as one-object categories. Then a functor has to send the unique object of to the unique object of . The action on morphisms is more interesting: from the definition of a functor, we see that a map of the underlying sets determines a functor if and only if it preserves the monoid operation and the identity, i.e. if and only if it is a monoid homomorphisms. Thus for monoids (and hence for groups), we can regard homomorphisms as functors.

Example C2.4 Let be a group, considered as a one-object category and let us consider functors to the category of sets (cf. C1.2), . Since has one proxy object , the value of corresponds to a choice of a particular set . As for the morphisms, we have for each a map such that and . These are precisely the axioms for a group action on , such that functors correspond to sets with a (left) -action, i.e. -Sets.

Example C2.5 By reasoning as in the last example, one can show that if is a field and we consider the category of vector spaces over with linear maps as morphisms, then functors correspond to representations of on vector spaces over .

Now we shall look into examples of particular functors.

Example C2.6 There are a plethora of so-called “forgetful” functors. These arise when you “forget” part of the structure of an object. In this case, the target category consists of objects with less structure than the source category.
For example, we can forget about the multiplication in a ring and are left with an abelian group, this gives rise to a forgetful functor from the category of rings to the category of abelian groups. We can then proceed on our path to oblivion and forget that an abelian group is abelian, yielding a forgetful functor from abelian groups to groups. Finally, every group is a set, giving rise to yet another forgetful functor. It should be noted that it is important to consider also the morphisms for these forgetful functors to work: every ring homomorphism is a homomorphism of abelian groups on the additive groups, every homomorphism of abelian groups is a group homomorphism and every group homomorphism is a map of sets. Another chain of forgetful functors goes from smooth complex varieties to complex manifolds, from complex manifolds to smooth manifolds, from smooth manifolds to topological manifolds, from topological manifolds to topological spaces and from topological spaces to sets, ending another journey of obliviousness in the same destination.
Forgetful functors are often implicitly applied, especially if one is not doing category theory. And indeed, it should deserve no particular mention that for instance, any group is a set. It can be quite helpful, however, to explicitly state when you apply them, for example to clarify in which category two objects are isomorphic or if you want to apply concepts or theorems on functors to a forgetful functor.

Example C2.7 One could say that the first example of functoriality is encountered in an introductory calculus class, although that would be a bit of a stretch.
Let be the category of pointed smooth manifolds, i.e. objects are pairs where is a smooth manifold and and morphisms are smooth maps such that . Then there’s the tangent space functor to the category of -vector spaces with linear maps that sends a pair to the tanget space of at and a morphism to the differential . The fact that this behaves functorially is an abstract version of the chain rule: , justifying the assertion on functoriality in calculus above.

Example C2.8 Let be the category of rings with ring homomorphisms as morphisms, then for any ring , the units form a group and ring homomorphisms preserve units so that a ring homomorphism restricts to a group homomorphism . We obtain a functor from to , the category of groups with group homomorphisms as morphisms.

Example C2.9 Let be an abelian group and , then there’s a functor from the category of topological spaces to the category of abelian groups that sends a space to the singular homology group . The fact that this is a functor is even part of the Eilenberg-Steenrod axioms for homology theories.

Example C2.10 Let be the category of topological spaces with continuous maps and let be the category of measurable spaces with measurable maps as morphisms. There’s a functor that equips a topological space with its Borel -algebra, turning it into a measurable space. Every continuous map is measurable with respect to the corresponding Borel -algebras, ensuring that this construction yields a functor.

Example C2.11 Not every constructions that seems in some sense “natural” can be turned into a functor. For example, we can associate to each group its center which is an abelian group. One could hope that this can be made into a functor. The problem is that central elements don’t necessarily get sent to central elements under a group homomorphism. Indeed, it is impossible to define any action on morphisms such that the assignment on objects becomes a functor, for the identity on the cyclic group on two elements factors through the inclusion of to as a transposition and the sign homomorphism . We have . However, the center of is itself, whereas the center of is trivial. If there was any functor that restricts to on objects, then we would get implying that the identity on factors over the trivial group, which is impossible.

Example C2.12 Let be a category and let , then for any object , we have a set and for a morphism , we can postcompose with , yielding a map . By associativity of composition and the definition of the identity elements, this construction yields a functor . These functors – affectionally called Hom-functors – are of paramount importance, as we shall see in future posts.

Example C2.13 We have said that functors can be thought of as morphisms between categories. This can be taken quite literally. To avoid Russel’s paradox and to ensure that our sets of morphisms will actually be sets, let’s restrict our attention to categories whose class of objects is actually a set, so-called “small” categories. There’s a (non-small!) category , in which the objects are small categories and morphisms are functors between small categories. Note that we have implicitly said that one can compose functors and that there’s an identity functor for each category.

Contravariance

All the previous examples of functors had one thing in common: they didn’t reverse the orientation of the arrows. Even so, there are other instances of functoriality. The following construction allows us to subsume those instances under the previous definition:

Definition C2.15 Let be a category, then the opposite category has the same objects. For , we set composition is defined like the composition in except in the opposite order.

Intuitively, we just reverse all the arrows.

Definition C2.16 Let and be categories, then a contravariant functor is a covariant functor .

For clarification, regular functors (as in definition C2.1) are called covariant, but if “functor” is used without specification, they are always covariant by default.

From this definition, it follows that a contravariant functor reverses the direction of morphisms when it is applied. (Hence the picture in the beginning of this section)

Example C2.16 Consider the category of sets . The assignment on objects can be made into a functor in two different ways, a covariant one and a contravariant one. For the covariant one, let be any map. Then we can define a map via , giving rise to the covariant powerset functor. For the contravariant one, we can define for a map a map via . In both cases the verification of the functorial properties is easy.

Example C2.17 Let be a field. Consider the category of finite separable extensions of with -linear field homomorphisms as morphisms. We can define a contravariant functor from to , the category of groups with group homomorphisms as morphisms, in the following manner: For an object , set and for a (necessarily injective) morphism , define to be the field norm . The transitivity formula for norms in towers translates into functoriality of this construction: For a tower of extensions we have .
A similar construction works for the field trace.

Example C2.18 Consider the category of compact Hausdorff spaces with continuous maps as morphisms. For any compact Hausdorff space , the continuous functions form a C*-algebra with pointwise operations (i.e. addition, multiplication, scalar multiplication and conjugation), denoted by . If and are compact Hausdorff spaces and is a continuous map, then we can define a homomorphism of C*-algebras by pulling back continuous functions along : for a continuous function , is a continuous function . We obtain a contravariant functor from the category of compact Hausdorff spaces to the category of commutative unital C*-algebras.

Example C2.19 As in the examples C2.4 and C2.5, if is a group, considered as a one-object category and we let be the category of sets or the category of -vector spaces for a field , then contravariant functors correspond to right -sets or right representations on vector spaces over , respectively.

Example C2.20 The construction from C2.12 has a contravariant sibling. Let be any category and . Then we have for any object in , a set and for a morphism , we get a map . This construction is a functor, called the contravariant Hom functor.

The purpose of this post is to establish some basic language of categories. This can be read independently of the ongoing series on representation theory, although concepts and results from this post will be used in the representation theory series. So with respect to the representation theory series, one can think of this post (and the follow-up posts) as the analog of an appendix.

The notions of category theory can be encountered in many different mathematical contexts. To illustrate the concepts, a range of examples will be given, but I will go quickly over standard examples – For common examples in greater detail and multitude, see the references below. It is not necessary to know the background of every single example to grasp the concepts.

To readers who wish to read a more detailed introduction, I recommend Tom Leinster’s “Basic Category Theory”  (available as a free pdf, but also published in book form) or in case you know German Martin Brandenburg’s “Einführung in die Kategorientheorie”. If you simply cannot get enough, Francis Bourceux’ three-volume Handbook of Categorical Algebra is the right place to get lost in the realm of categories and functors.

The views expressed on category theory are naturally (no pun intended) my own highly subjective ones and others may differ, especially those with a categorical disapproval of categorical things.

Introduction

The meta-mathematical notions of category theory provide a unifying framework for a lot of different mathematical theories. (Merely having a framework, however, can’t always replace delving into the specifics of the particular objects that one is interested in.) We can not only translate statements into categorical language, but we can also use categorical language to make things precise that might otherwise be vague, e.g. what it means for two different kinds of mathematical structures to be “equivalent” or what it means for a map to be “natural”.

The basic objects of category theory are categories. One should think of a category as a big directed graph in which objects are represented by nodes and morphisms are represented by arrows and where we have a special operation called composition that takes two consecutive arrows and produces a third arrow representing going along one arrow first and then along the other one.
Compared with set theory, which can be called “materialistic”, in the sense that it is about what objects actually are made out of, category theory can be considered “behavioristic”, in the sense that it’s more about how objects behave, i.e. relate to each other. Relations between objects are conceptualized by the arrows also known as morphisms. But for morphisms, just like for the objects themselves, it’s more about how they relate each other and less about what they actually represent. The relations between morphisms are given by composition.

Thinking categorically, we can economize on routine verifications by doing them in the utmost generality. More importantly, category theory can expose formal similarities between seemingly disconnected phenomena. These analogies might only become apparent through the high level of abstraction category-theoretic concepts provide, allowing one to transfer intuition, techniques and concepts in ways not available otherwise, at least not as easily or as precisely-defined.

Peter Freyd wrote once that “Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial.”
Peter May commented: “I prefer an update of that quote: ‘ Perhaps the purpose of categorical algebra is to show that which is formal is formally formal’. It is by now abundantly clear that mathematics can be formal without being trivial.”

Now then! After this informal introduction, let us delve into the formal details of why that which is formal is formally formal.

Categories

Definition C1.1 A category consists of the following data:

• A class of objects
• For every pair of objects , a set of morphisms
• For every triple of objects , a map called composition and denoted by

Such that the following conditions hold:

• For any object , there exists an identity such that for all objects and all morphisms we have and
• Composition is associative, that means for all objects and , we have
• unless and

Remark The last condition is just a technical one that is not very interesting and is mostly omitted from verifications that things are categories. You can always replace the Hom-sets by disjoint copies by set-theoretic juggling, so it’s not much of an issue. The advantage of this convention is that source and target of a morphism are uniquely determined.

Example C1.2 We can form the category of sets , where is the class of all sets and for , is the set of all maps from to and the composition of morphisms is just the usual composition of maps. Many other examples (but not all of them) can be formed as “subcategories” of this example in the sense that the objects are sets with additional structure and morphisms are mappings which preserve that structure in some way, such as topological spaces and continuous maps or algebraic structures of a specific type and homomorphisms or smooth manifolds and smooth maps or complex manifolds and holomorphic maps or varieties over an algebraically closed fields and regular maps etc.

Example C1.3 There is another category with the class of all sets as objects, but with more morphisms. Recall that a mapping is a particular kind of relation. is the category consisting of all sets as objects, and for two objects , is the set of all relations from to , i.e. subsets . Given three sets and relations , we can form the composite relation . (If and are functions, this reduces to regular function composition.) With this composition, satisfies the axioms for a category.

Example C1.4 In the previous examples for categories, the objects were given by the class of all sets. But there’s also the possibility to fix a particular set and consider as a category in the following way: and the only morphisms are the identity morphisms required by the definition of categories. Due to the striking similarity with discrete topological spaces (compare for instance the set of morphisms if and if with the discrete metric), this construction is called the discrete category on the set .

Example C1.5 Instead of only allowing for the minimum amount of morphisms, we can take a set and consider such categories with as their class of objects that have at most one morphism between two objects. In such a case, the only interesting information is whether there is a morphism between two objects or not. Consequently, such a category corresponds to a relation . But the requirements for compositions and identities force a relation to have some properties if it is supposed to define a category in this manner: the existence of identity morphisms in the supposed category is equivalent to the relation being reflexive and the existence of compositions translates to transivity. A reflexive and transitive relation is called “preorder”. For instance, all equivalence relations and all partial orderings are preorders. An example of a preorder that is neither can be formed by taking a ring, say, an integral domain such as and considering the divisibility relation on elements of . Here the failure to be a partial order is due to the existence of non-trivial units and the failure to be an equivalence relation is to due the existence of non-invertible elements.

Definition C1.6 Speaking of invertible things, just like for an element in a ring, we can define when a morphism in a category is invertible. Let a morphism in a category. Then is called invertible, or an isomorphism, if there is a morphism such that and . If there is such a morphism, then and are called isomorphic. Note that the notations and , albeit reminiscent of the situation in the category , don’t necessarily refer to mappings or function composition, but to the particular morphisms and composition in an arbitrary category.

Example C1.7 Other than restricting the size of morphisms as in example C1.5, one can also restrict the size of the class of objects. For this, let’s consider categories with only one object. In general, if we consider composition as a function on all morphisms of a category, it is only partially defined, since source and target need to match up correctly. But this doesn’t arise if we have only one object. Let be a category with one object and let . Then composition is a (totally defined) mapping , i.e. a binary operation.
The existence of the identity implies that is a unit with respect to this binary operation and the associativity of the composition implies that the binary operation is, well, associative. Thus composition makes into a monoid.
Conversely, given a monoid , we can turn into a one-object category by taking a proxy object with no particular meaning, setting and defining the composition via the monoid operation. These constructions are inverse to each other, such that monoids correspond to one-object categories.
A particular frequently occuring and much beloved class of monoids consists of groups, i.e. monoids in which every element is invertible. By a pleasant convergence of terminology, an element in a monoid being invertible (in the classical sense) is the same as being an invertible morphism if we consider the monoid as a one-object category. Thus we can say that a monoid is a group if and only if every morphism is an isomorphism. This leads to the following notion:

Groupoids

Definition C1.8 A groupoid is a category such that every morphism is an isomorphism.

If we think of groups as encoding the symmetries of one object, then groupoids can capture the symmetries of a configuration with many objects.

Example C1.9 Let be a preorder on a set , then we can form a category from these data as described in example C1.8. One can ask when one obtains groupoid in this way. It turns out that the corresponding category is a groupoid if and only if is symmetric, i.e. an equivalence relation.

Remark From this observation we can extract an intuition for general groupoids: a groupoid is like a set with an equivalence relation, except that two objects can be equivalent in more than one way and we’re keeping track of different ways of being equivalent. From this viewpoint a group, being a one-object groupoid, is like the various ways a particular object is equivalent to itself, following the narrative that groups encode the symmetries of an object.

Example C1.10 Let be a group acting on a set , then one can form a groupoid encoding the group action as follows: Let be the category having elements from as its objects and for every and , there’s a morphism , such that is in bijection with . This groupoid played an important role in a previous post.

Example C1.11 Let be any category. Then there is a maximal subcategory that is a groupoid: simply take the subcategory of with the same objects as and all the isomorphisms of as morphisms. This is called the core of the category and it is a groupoid. If has only one object, i.e. is a monoid, then the core of is just the group of invertible elements, also known as units.
As an exercise, one can check that the categories and (cf. C1.2 and C1.3) have the same core, i.e. any invertible relation is already a function and in fact a bijection.

Example C1.12 Let be a topological space. Then the fundamental groupoid has as its objects all elements of . For , consists of all homotopy classes of paths from to . Composition is given by concatenation of paths. contains important homotopy-theoretic information about , such as the path-components and the fundamental group of each path-component. For more information on this very geometric construction and groupoids in algebraic topology, I heartily recommend Ronald Brown’s “Topology and Groupoids”.

This concludes the first part of the mini-series on introductory categories. Next time we’ll look into functors.