Category theory has come to occupy a central position in contemporarymathematics and theoretical computer science, and is also applied tomathematical physics. Roughly, it is a general mathematical theory ofstructures and of systems of structures. As category theory is stillevolving, its functions are correspondingly developing, expanding andmultiplying. At minimum, it is a powerful language, or conceptualframework, allowing us to see the universal components of a family ofstructures of a given kind, and how structures of different kinds areinterrelated. Category theory is both an interesting object ofphilosophical study, and a potentially powerful formal tool forphilosophical investigations of concepts such as space, system, andeven truth. It can be applied to the study of logical systems in whichcase category theory is called “categorical doctrines” atthe syntactic, proof-theoretic, and semantic levels. Category theoryeven leads to a different theoretical conception of set and, as such,to a possible alternative to the standard set theoretical foundationfor mathematics. As such, it raises many issues about mathematicalontology and epistemology. Category theory thus affords philosophersand logicians much to use and reflect upon.

• 1. General Definitions, Examples and Applications
  • 1.1 Definitions
  • 1.2 Examples
  • 1.3 Fundamental Concepts of the Theory
• 2. Brief Historical Sketch
• 3. Philosophical Significance
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. General Definitions, Examples and Applications

1.1 Definitions

Categories are algebraic structures with many complementary natures,e.g., geometric, logical, computational, combinatorial, just as groupsare many-faceted algebraic structures. Eilenberg & Mac Lane(1945) introduced categories in a purely auxiliary fashion, aspreparation for what they called functors and naturaltransformations. The very definition of a category evolved over time,according to the author’s chosen goals and metamathematicalframework. Eilenberg & Mac Lane at first gave a purelyabstract definition of a category, along the lines of the axiomaticdefinition of a group. Others, starting with Grothendieck (1957) andFreyd (1964), elected for reasons of practicality to define categoriesin set-theoretic terms.

An alternative approach, that of Lawvere (1963, 1966), begins bycharacterizing the category of categories, and then stipulates that acategory is an object of that universe. This approach, under activedevelopment by various mathematicians, logicians and mathematicalphysicists, lead to what are now called “higher-dimensionalcategories” (Baez 1997, Baez & Dolan 1998a, Batanin 1998, Leinster2002, Hermidaet al. 2000, 2001, 2002). The very definitionof a category is not without philosophical importance, since one ofthe objections to category theory as a foundational framework is theclaim that since categories aredefined as sets, categorytheory cannot provide a philosophically enlightening foundation formathematics. We will briefly go over some of these definitions,starting with Eilenberg’s & Mac Lane’s (1945) algebraicdefinition. However, before going any further, the followingdefinition will be required.

As Eilenberg & Mac Lane promptly remark, objects play asecondary role and could be entirely omitted from the definition.Doing so, however, would make the manipulation of the applicationsless convenient. It is practically suitable, and perhapspsychologically more simple to think in terms of mappings andobjects. The term “aggregate” is used by Eilenberg &Mac Lane themselves, presumably so as to remain neutral withrespect to the background set theory one wants to adopt.

Eilenberg & Mac Lane defined categories in 1945 for reasonsof rigor. As they note:

It should be observed first that the whole concept of a category isessentially an auxiliary one; our basic concepts are essentially thoseof a functor and of natural transformation (…). The idea of acategory is required only by the precept that every function shouldhave a definite class as domain and a definite class as range, for thecategories are provided as the domains and ranges of functors. Thusone could drop the category concept altogether and adopt an even moreintuitive standpoint, in which a functor such as “Hom” isnot defined over the category of “all” groups, but foreach particular pair of groups which may be given. The standpointwould suffice for applications, inasmuch as none of our developmentswill involve elaborate constructions on the categoriesthemselves. (1945, chap. 1, par. 6, p. 247)

Things changed in the following ten years, when categories startedto be used in algebraic topology and homological algebra. Mac Lane,Buchsbaum, Grothendieck and Heller were considering categories in whichthe collections of morphisms between two fixed objects have an additionalstructure. More specifically, given any two objects \(X\) and\(Y\) of a category \(\mathbf{C}\), theset\(\mathbf{Hom}(X, Y)\) of morphisms from\(X\) to \(Y\) form an abelian group. Furthermore, forreasons related to the ways homology and cohomology theories arelinked, the definition of a category had to satisfy an additionalformal property (which we will leave aside for the moment): it had tobe self-dual. These requirements lead to the following definition.

This is the definition one finds in most textbooks of categorytheory. As such it explicitly relies on a set theoretical backgroundand language. An alternative, suggested by Lawvere in the earlysixties, is to develop an adequate language and background frameworkfor a category of categories. We will not present the formal frameworkhere, for it would take us too far from our main concern, but thebasic idea is to define what are called weak \(n\)-categories(and weak \(\omega\)-categories), and what had been called categorieswould then be called weak 1-categories (and sets would be weak0-categories). (See, for instance, Baez 1997, Makkai 1998, Leinster2004, Baez & May 2010, Simpson 2011.)

Also in the sixties, Lambek proposed to look at categories asdeductive systems. This begins with the notion of agraph,consisting of two classesArrows andObjects, andtwo mappings between them, \(\boldsymbol{s}: \Arrows \rightarrow\Objects\) and \(\boldsymbol{t}: \Arrows \rightarrow \Objects\),namely the source and the target mappings. The arrows are usuallycalled the “oriented edges” and the objects“nodes” or “vertices”. Following this,adeductive system is a graph with a specified arrow:

(R1) \(\mathbf{id}_X : X \rightarrow X\),

and a binary operation on arrows:

(R2) Given \(\boldsymbol{f}: X \rightarrow Y\) and \(\boldsymbol{g}: Y \rightarrow Z\), the composition of \(\boldsymbol{f}\) and\(\boldsymbol{g}\) is \((\boldsymbol{g} \circ \boldsymbol{f}): X \rightarrow Z.\)

Of course, the objects of a deductive system are normally thought ofasformulas, the arrows are thought of asproofs ordeductions, and operations on arrows are thought of asrules of inference. Acategory is then definedthus:

Thus, by imposing an adequate equivalence relation upon proofs, anydeductive system can be turned into a category. It is thereforelegitimate to think of a category as an algebraic encoding of adeductive system. This phenomenon is already well-known to logicians,but probably not to its fullest extent. An example of such analgebraic encoding is the Lindenbaum-Tarski algebra, a Boolean algebracorresponding to classical propositional logic. Since a Booleanalgebra is a poset, it is also a category. (Notice also that Booleanalgebras with appropriate homomorphisms between them form anotheruseful category in logic.) Thus far we have merely a change ofvocabulary. Things become more interesting when first-order andhigher-order logics are considered. The Lindenbaum-Tarski algebra forthese systems, when properly carried out, yields categories, sometimescalled “conceptual categories” or “syntactic categories”(Mac Lane & Moerdijk 1992, Makkai & Reyes 1977, Pitts2000).

1.2 Examples

Almost every known example of a mathematical structure with theappropriate structure-preserving map yields a category.

1. The categorySet with objects sets and morphisms the usualfunctions. There are variants here: one can consider partial functionsinstead, or injective functions or again surjective functions. In eachcase, the category thus constructed is different.
2. The categoryTop with objects topological spaces andmorphisms continuous functions. Again, one could restrict morphisms toopen continuous functions and obtain a differentcategory.
3. The categoryhoTop with objects topological spaces andmorphisms equivalence classes of homotopic functions. This category isnot only important in mathematical practice, it is at the core ofalgebraic topology, but it is also a fundamental example of a categoryin which morphisms arenot structure preservingfunctions.
4. The categoryVec with objects vector spaces and morphismslinear maps.
5. The categoryDiff with objects differential manifolds andmorphisms smooth maps.
6. The categoriesPord andPoSet with objectspreorders and posets, respectively, and morphisms monotonefunctions.
7. The categoriesLat andBool with objects latticesand Boolean algebras, respectively, and morphisms structure preservinghomomorphisms, i.e., \((\top , \bot , \wedge , \vee)\) homomorphisms.
8. The categoryHeyt with objects Heyting algebras and \((\top , \bot , \wedge , \vee , \rightarrow)\) homomorphisms.
9. The categoryMon with objects monoids and morphisms monoidhomomorphisms.
10. The categoryAbGrp with objects abelian groups andmorphisms group homomorphisms, i.e. \((1, \times, (-)^{-1})\) homomorphisms.
11. The categoryGrp with objects groups and morphisms grouphomomorphisms, i.e. \((1, \times, (-)^{-1})\) homomorphisms.
12. The categoryRings with objects rings (with unit) andmorphisms ring homomorphisms, i.e. \((0, 1, +, \times)\)homomorphisms.
13. The categoryFields with objects fields and morphismsfields homomorphisms, i.e. \((0, 1, +, \times)\) homomorphisms.
14. Any deductive system \(T\) with objects formulae and morphismsproofs.

These examples nicely illustrates how category theory treats thenotion of structure in a uniform manner. Note that a category ischaracterized by its morphisms, and not by its objects. Thus thecategory of topological spaces with open maps differs from thecategory of topological spaces with continuous maps — or, moreto the point, the categorical properties of the latter differ fromthose of the former.

We should underline again the fact that not all categories are madeof structured sets with structure-preserving maps. Thus any preorderedset is a category. For given two elements \(p, q\) of apreordered set, there is a morphism\(\boldsymbol{f}: p \rightarrow q\) if and only if\(p \le q.\) Hence a preordered set is a category in which thereis at most one morphism between any two objects. Any monoid (and thusany group) can be seen as a category: in this case the category hasonly one object, and its morphisms are the elements of the monoid.Composition of morphisms corresponds to multiplication of monoidelements. That the monoid axioms correspond to the category axioms iseasily verified.

Hence the notion of category generalizes those of preorder andmonoid. We should also point out that a groupoid has a very simpledefinition in a categorical context: it is a category in which everymorphism is an isomorphism, that is for any morphism\(\boldsymbol{f}: X \rightarrow Y\),there is a morphism \(\boldsymbol{g}: Y \rightarrow X\) such that\(\boldsymbol{f} \circ \boldsymbol{g} = \mathbf{id}_X\) and\(\boldsymbol{g} \circ \boldsymbol{f} = \mathbf{id}_Y.\)

1.3 Fundamental Concepts of the Theory

Category theory unifies mathematical structures in two different ways.First, as we have seen, almost every set theoretically definedmathematical structure with the appropriate notion of homomorphismyields a category. This is a unification providedwithin a settheoretical environment. Second, and perhaps even more important, oncea type of structure has been defined, it is imperative to determine hownew structures can be constructed out of the given one. For instance,given two sets \(A\) and \(B\), set theory allows us toconstruct their Cartesian product \(A \times B.\) It is alsoimperative to determine how given structures can be decomposed intomore elementary substructures. For example, given a finite Abeliangroup, how can it be decomposed into a product of certain of itssubgroups? In both cases, it is necessary to know how structures of acertain kind may combine. The nature of these combinations might appearto be considerably different when looked at from a purely settheoretical perspective.

Category theory reveals that many of these constructions are in factcertain objects in a category having a “universal property”. Indeed,from a categorical point of view, a Cartesian product in set theory, adirect product of groups (Abelian or otherwise), a product oftopological spaces, and a conjunction of propositions in a deductivesystem are all instances of a categorical product characterized by auniversal property. Formally, aproduct of two objects\(X\) and \(Y\) in a category \(\mathbf{C}\) is an object\(Z\) of \(\mathbf{C}\)together with two morphisms,called the projections, \(\boldsymbol{p}: Z \rightarrow X\) and \(\boldsymbol{q}: Z \rightarrow Y\)such that—and this is the universal property—for allobjects \(W\) with morphisms \(\boldsymbol{f}: W \rightarrow X\)and \(\boldsymbol{g}: W \rightarrow Y\),there is a unique morphism\(\boldsymbol{h}: W \rightarrow Z\) such that\(\boldsymbol{p} \circ \boldsymbol{h} = \boldsymbol{f}\) and \(\boldsymbol{q} \circ \boldsymbol{h} = \boldsymbol{g}.\)

Note that we have defined \(a\) product for \(X\) and\(Y\) and notthe product for \(X\) and \(Y.\)Indeed, products and other objects with a universal property aredefined only up to a (unique) isomorphism. Thus in category theory,the nature of the elements constituting a certain construction isirrelevant. What matters is the way an object is related to the otherobjects of the category, that is, the morphisms going in and themorphisms going out, or, put differently, how certain structures canbe mapped into a given object and how a given object can map itsstructure into other structures of the same kind.

Category theory reveals how different kinds of structures arerelated to one another. For instance, in algebraic topology,topological spaces are related to groups (and modules, rings, etc.) invarious ways (such as homology, cohomology, homotopy, K-theory). As notedabove, groups with group homomorphisms constitute a category. Eilenberg& Mac Lane invented category theory precisely in order toclarify and compare these connections. What matters are the morphismsbetween categories, given by functors. Informally, functors arestructure-preserving maps between categories. Given two categories\(\mathbf{C}\) and \(\mathbf{D}\), a functor \(F\) from\(\mathbf{C}\) to \(\mathbf{D}\) sends objects of\(\mathbf{C}\) to objects of \(\mathbf{D}\), and morphisms of\(\mathbf{C}\) to morphisms of \(\mathbf{D}\), in such a waythat composition of morphisms in \(\mathbf{C}\) is preserved, i.e.,\(F(\boldsymbol{g} \circ \boldsymbol{f}) = F(\boldsymbol{g}) \circ F(\boldsymbol{f})\),and identity morphisms are preserved, i.e.,\(F(\mathbf{id}_X ) = \mathbf{id}_{FX}.\) It immediately follows thata functor preserves commutativity of diagrams between categories.Homology, cohomology, homotopy, K-theory are all example offunctors.

A more direct example is provided by the power set operation, whichyields two functors on the category of sets, depending on how onedefines its action on functions. Thus given a set \(X,\wp(X)\) is the usual set of subsets of \(X\), andgiven a function \(\boldsymbol{f}: X \rightarrow Y, \wp(\boldsymbol{f}):\wp(X) \rightarrow \wp(Y)\) takes a subset\(A\) of \(X\) and maps it to \(B = \boldsymbol{f}(A)\), the image of\(\boldsymbol{f}\) restricted to \(A\) in \(X.\) Itis easily verified that this defines a functor from the category ofsets into itself.

In general, there are many functors between two given categories, andthe question of how they are connected suggests itself. For instance, given acategory \(\mathbf{C}\), there is always the identity functor from\(\mathbf{C}\) to \(\mathbf{C}\) which sends everyobject/morphism of \(\mathbf{C}\) to itself. In particular, thereis the identity functor over the category of sets.

Now, the identity functor is related in a natural manner to the power set functordescribed above. Indeed, given a set \(X\) andits power set \(\wp(X)\), there is a function\(\mathbf{h}_X\) which takes an element\(x\) of \(X\) and sends it to the singleton set \(\{x\}\),a subset of \(X\), i.e., an element of \(\wp(X).\) Thisfunction in fact belongs to a family of functions indexed by theobjects of the category of sets\(\{\mathbf{h}_Y : Y \rightarrow \wp(X)| Y \text{ in } \mathbf{Ob(Set)}\}.\)Moreover, it satisfies the following commutativity condition: given anyfunction \(\boldsymbol{f}: X \rightarrow Y\), theidentity functor yields the same function\(\boldsymbol{Id}(\boldsymbol{f}):\boldsymbol{Id}(X) \rightarrow \boldsymbol{Id}(Y).\) The commutativity conditionthus becomes: \(\mathbf{h}_Y \circ \boldsymbol{Id}(\boldsymbol{f}) = \wp(\boldsymbol{f}) \circ \mathbf{h}_X.\)Thus the family of functions \(\boldsymbol{h}({\text{-}})\) relates thetwo functors in a natural manner. Such families of morphisms are callednatural transformations between functors. Similarly, naturaltransformations between models of a theory yield the usualhomomorphisms of structures in the traditional set theoreticalframework.

The above notions, while important, are not fundamental to categorytheory. The latter heading arguably include the notions oflimit/colimit; in turn, these are special cases of what is certainlythe cornerstone of category theory, the concept of adjoint functors,first defined by Daniel Kan in 1956 and published in 1958.

Adjoint functors can be thought of as being conceptual inverses.This is probably best illustrated by an example. Let\(U: \mathbf{Grp} \rightarrow \mathbf{Set}\)be the forgetful functor, that is, the functor that sends to eachgroup \(G\) its underlying set ofelements \(U(G)\), and to a grouphomomorphism \(\boldsymbol{f}: G \rightarrow H\) the underlying set function\(U(\boldsymbol{f}): U(G) \rightarrow U(H).\) In other words, \(U\) forgets about thegroup structure and forgets the fact that morphisms are grouphomomorphisms. The categories \(\mathbf{Grp}\) and\(\mathbf{Set}\) are certainly not isomorphic, as categories, toone another. (A simple argument runs as follows: the category\(\mathbf{Grp}\) has a zero object, whereas \(\mathbf{Set}\)does not.) Thus, we certainly cannot find an inverse, in the usualalgebraic sense, to the functor \(U.\) But there are manynon-isomorphic ways to define a group structure on a given set\(X\), and one might hope that among these constructions at leastone is functorial and systematically related to the functor \(U.\)What is the conceptual inverse to the operation of forgetting all thegroup theoretical structure and obtaining a set? It is to construct agroup from a set solely on the basis of the concept of group andnothing else, i.e., with no extraneous relation or data. Such a group isconstructed “freely”; that is, with no restriction whatsoever exceptthose imposed by the axioms of the theory. In other words, all that isremembered in the process of constructing a group from a given set isthe fact that the resulting construction has to be a group. Such aconstruction exists; it is functorial and it yields what are calledfree groups. In other words, there is a functor\(F: \mathbf{Set} \rightarrow \mathbf{Grp}\),which to any set \(X\) assigns the free group\(F(X)\) on \(X\), and to each function\(\boldsymbol{f}: X \rightarrow Y\), the grouphomomorphism \(F(\boldsymbol{f}): F(X) \rightarrow F(Y)\), defined in theobvious manner. The situation can be described thusly: we have twoconceptual contexts, a group theoretical context and a set theoreticalcontext, and two functors moving systematically from one context to theother in opposite directions. One of these functors is elementary,namely the forgetful functor \(U.\) It is apparently trivial anduninformative. The other functor is mathematically significant andimportant. The surprising fact is that \(F\) is related to\(U\) by a simple rule and, in some sense, it arises from\(U.\) One of the striking features of adjoint situations isprecisely the fact that fundamental mathematical and logicalconstructions arise out of given and often elementary functors.

The fact that \(U\) and \(F\) are conceptual inversesexpresses itself formally as follows: applying \(F\) first andthen \(U\) does not yield the original set \(X\), but thereis a fundamental relationship between \(X\) and\(UF(X).\) Indeed, there is a function \(\eta\):\(X \rightarrow UF(X)\), called theunit of theadjunction, that simply sends each element of \(X\) to itselfin \(UF(X)\) and this function satisfies the followinguniversal property: given any function\(\boldsymbol{g}: X \rightarrow U(G)\), there is a uniquegroup homomorphism\(\boldsymbol{h}: F(X) \rightarrow G\) such that \(U(\boldsymbol{h}) \circ \eta = \boldsymbol{g}.\) In other words, \(UF(X)\) isthe best possible solution to the problem of inserting elements of\(X\) into a group (what is called “insertion of generators” inthe mathematical jargon). Composing \(U\) and \(F\) in theopposite order, we get a morphism \(\xi : FU(G)\rightarrow G\), called thecounit of the adjunction,satisfying the following universal property: for any group homomorphism\(\boldsymbol{g}: F(X) \rightarrow G\), there is a unique function\(\boldsymbol{h}: X \rightarrow U(G)\) such that \(\xi \circF(\boldsymbol{h}) = \boldsymbol{g}\). In other words, \(FU(G)\)constitutes the best possible solution to the problem of finding arepresentation of \(G\) as a quotient of a free group. If \(U\) and\(F\) were simple algebraic inverses to one another, we would have thefollowing identity: \(UF = I_{\mathbf{Set}}\) and \(FU =I_{\mathbf{Grp}}\), where \(I_{\mathbf{Set}}\) denotes the identityfunctor on \(\mathbf{Set}\) and \(I_{\mathbf{Grp}}\) the identityfunctor on \(\mathbf{Grp}.\) As we have indicated, these identitiescertainly do not hold in this case. However, some identities do hold:they are best expressed with the help of the commutative diagrams:

where the diagonal arrows denote the appropriate identity naturaltransformations.

This is but one case of a very common situation: every freeconstruction can be described as arising from an appropriate forgetfulfunctor between two adequately chosen categories. The number ofmathematical constructions that can be described as adjoints is simplystunning. Although the details of each one of these constructions varyconsiderably, the fact that they can all be described using the samelanguage illustrates the profound unity of mathematical concepts andmathematical thinking. Before we give more examples, a formal andabstract definition of adjoint functors is in order.

Here are some of the important facts regarding adjoint functors.Firstly, adjoints are unique up to isomorphism; that is any two leftadjoints \(F\) and \(F'\) of a functor \(G\) arenaturally isomorphic. Secondly, the notion of adjointness is formallyequivalent to the notion of a universal morphism (or construction) andto that of representable functor. (See, for instance Mac Lane1998, chap. IV.) Each and every one of these notions exhibit anaspect of a given situation. Thirdly, a left adjoint preserves all thecolimits which exist in its domain, and, dually, a right adjointpreserves all the limits which exist in its domain.

We now give some examples of adjoint situations to illustrate thepervasiveness of the notion.

1. Instead of having a forgetful functor going into the category ofsets, in some cases only a part of the structure is forgotten. Here aretwo standard examples:
   • There is an obvious forgetful functor \(U\):AbGrp \(\rightarrow\)AbMon from the category of abeliangroups to the category of abelian monoids: \(U\) forgets aboutthe inverse operation. The functor \(U\) has a left adjoint\(F\):AbMon \(\rightarrow\)AbGrp which, given anabelian monoid \(M\), assigns to it the best possible abeliangroup \(F(M)\) such that \(M\) can be embedded in\(F(M)\) as a submonoid. For instance, if \(M\)is \(\mathbb{N}\), then \(F(\mathbb{N})\) “is” \(\mathbb{Z}\), that is, itis isomorphic to \(\mathbb{Z}.\)
   • Similarly, there is an obvious forgetful functor\(U: \mathbf{Haus} \rightarrow \mathbf{Top}\)from the category of Hausdorff topological spaces to the category oftopological spaces which forgets the Hausdorff condition. Again, thereis a functor\(F: \mathbf{Top} \rightarrow \mathbf{Haus}\) such that\(F \dashv U.\) Given a topological space \(X,F(X)\) yields the best Hausdorff space constructed from\(X\): it is the quotient of \(X\) by the closure of thediagonal \(\overline{\Delta}_X \subseteq X \times X\), which is an equivalence relation. Incontrast with the previous example where we had an embedding, thistime we get a quotient of the original structure.
2. Consider now the category ofcompact Hausdorff spaceskHaus and the forgetful functor \(U\):kHaus \(\rightarrow \mathbf{Top}\), which forgets the compactnessproperty and the separation property. The left adjoint to this\(U\) is the Stone-Cech compactification.
3. There is a forgetful functor \(U:\textbf{Mod}_R \rightarrow\)AbGrp from a category of \(R\)-modules to thecategory of abelian groups, where \(R\) is a commutative ringwith unit. The functor \(U\) forgets the action of \(R\) ona group \(G.\) The functor \(U\) has both a left and a rightadjoint. The left adjoint is \(R \otimes -\):AbGrp \(\rightarrow \mathbf{Mod}_R\) which sends an abelian group\(G\) to the tensor product \(R \otimes G\) and the right adjoint is given by the functor\(\mathbf{Hom}(R, -)\):AbGrp\(\rightarrow \mathbf{Mod}_R\) which assigns to any group\(G\) the modules of linear mappings\(\mathbf{Hom}(R, G).\)
4. The case where the categories \(\mathbf{C}\) and\(\mathbf{D}\) are posets deserves special attention here. Adjointfunctors in this context are usually calledGaloisconnections. Let \(\mathbf{C}\) be a poset. Consider thediagonal functor \(\Delta : \mathbf{C} \rightarrow \mathbf{C} \times \mathbf{C}\), with \(\Delta(X) = \langle X,X\rangle\) and for \(\boldsymbol{f}:X \rightarrow Y, \Delta(\boldsymbol{f}) = \langle \boldsymbol{f},\boldsymbol{f}\rangle : \langle X,X\rangle \rightarrow \langle Y, Y\rangle.\) Inthis case, the left-adjoint to \(\Delta\) is the coproduct, or thesup, and the right-adjoint to \(\Delta\) is the product, or the inf.The adjoint situation can be described in the following special form: \[\frac{X \vee Y \le Z}{X \le Z, Y \le Z}\Updownarrow \qquad\frac{Z \le X \wedge Y}{Z \le Y, Z \le X}\Updownarrow\]

   where the vertical double arrow can be interpreted as rules ofinference going in both directions.

5. Implication can also be introduced. Consider a functor with aparameter: \((- \wedge X): \mathbf{C} \rightarrow \mathbf{C}.\) Itcan easily be verified that when \(\mathbf{C}\) is a poset, thefunction \((- \wedge X)\) is order preserving and thereforea functor. A right adjoint to \((- \wedge X)\) is a functorthat yields the largest element of \(\mathbf{C}\) such that itsinfimum with \(X\) is smaller than \(Z.\) This element issometimes called the relative pseudocomplement of \(X\) or, morecommonly, theimplication. It is denoted byX \(\Rightarrow\)Z or by\(X \supset Z.\) The adjunction can be presented as follows:\[\frac{Y \wedge X \le Z}{Y \le X \Rightarrow Z}\Updownarrow\]
6. The negation operator \(\neg X\) can be introduced from thelast adjunction. Indeed, let \(Z\) be the bottom element \(\bot\) ofthe lattice. Then, since \(Y \wedge X \le \bot\) is always true, it follows that\(Y \le X \Rightarrow \bot\) is also always true. But since \(X \Rightarrow \bot \le X\) is always the case, we get at the numeratorthat \((X \Rightarrow \bot \wedge X) = \bot.\) Hence, \(X \Rightarrow \bot\) isthe largest element disjoint from \(X.\) We can therefore put\(\neg X =_{def} X \Rightarrow \bot.\)
7. Limits, colimits, and all the fundamental constructions ofcategory theory can be described as adjoints. Thus, products andcoproducts are adjoints, as are equalizers, coequalizers, pullbacksand pushouts, etc. This is one of the reasons adjointness is centralto category theory itself: because all the fundamental operations ofcategory theory arise from adjoint situations.
8. Anequivalence of categories is a special case ofadjointness. Indeed, if in the above triangular identities the arrows\(\eta : I_{\mathbf{C}} \rightarrow GF\) and \(\xi : FG \rightarrow I_{\mathbf{D}}\) are naturalisomorphisms, then the functors \(F\) and \(G\)constitute anequivalence of categories. In practice, it isthe notion of equivalence of categories that matters and not thenotion of isomorphism of categories.

It is easy to prove certain facts about these operations directly fromthe adjunctions. Consider, for instance, implication. Let \(Z = X.\) Then we get at the numerator that \(Y \wedge X \le X\),which is always true in a poset (as is easily verified). Hence,\(Y \le X \Rightarrow X\) is also true forall \(Y\) and this is only possible if \(X \Rightarrow X = \top\), the top element of the lattice. Not onlycan logical operations be described as adjoints, but they naturallyarise as adjoints to basic operations. In fact, adjoints can be usedto define various structures, distributive lattices, Heyting algebras,Boolean algebras, etc. (See Wood, 2004.) It should be clear from thesimple foregoing example how the formalism of adjointness can be usedto give syntactic presentations of various logicaltheories. Furthermore, and this is a key element, the standarduniversal and existential quantifiers can be shown to be arising asadjoints to the operation of substitution. Thus, quantifiers are on apar with the other logical operations, in sharp contrast with theother algebraic approaches to logic. (See, for instance Awodey 1996 orMac Lane & Moerdijk 1992.) More generally, Lawvere showedhow syntax and semantics are related by adjoint functors. (See Lawvere1969b.)

Dualities play an important role in mathematics and they can bedescribed with the help of equivalences between categories. In otherwords, many important mathematical theorems can be translated asstatements about the existence of adjoint functors, sometimessatisfying additional properties. This is sometimes taken asexpressing theconceptual content of the theorem. Considerthe following fundamental case: let \(\mathbf{C}\) be the categorywhose objects are the locally compact abelian groups and the morphismsare the continuous group homomorphisms. Then, the Pontryagin dualitytheorem amounts to the claim that the category \(\mathbf{C}\) isequivalent to the category \(\mathbf{C}\)°, that is, to theopposite category. Of course, the precise statement requires that wedescribe the functors \(F: \mathbf{C} \rightarrow \mathbf{C}\)° and \(G: \mathbf{C}\)° \(\rightarrow \mathbf{C}\) and prove that they constitute an equivalence ofcategories.

Another well known and important duality was discovered by Stone inthe thirties and now bears his name. In one direction, an arbitraryBoolean algebra yields a topological space, and in the otherdirection, from a (compact Hausdorff and totally disconnected)topological space, one obtains a Boolean algebra. Moreover, thiscorrespondence is functorial: any Boolean homomorphism is sent to acontinuous map of topological spaces, and, conversely, any continuousmap between the spaces is sent to a Boolean homomorphism. In otherwords, there is an equivalence of categories between the category ofBoolean algebras and the dual of the category of Boolean spaces (alsocalled Stone spaces). (See Johnstone 1982 for an excellentintroduction and more developments.) The connection between a categoryof algebraic structures and the opposite of a category of topologicalstructures established by Stone’s theorem constitutes but one exampleof a general phenomenon that did attract and still attracts a greatdeal of attention from category theorists. Categorical study ofduality theorems is still a very active and significant field, and islargely inspired by Stone’s result. (For recent applications in logic,see, for instance Makkai 1987, Taylor 2000, 2002a, 2002b, Caramello 2011.)

2. Brief Historical Sketch

It is difficult to do justice to the short but intricate history of thefield. In particular it is not possible to mention all those who havecontributed to its rapid development. With this word of caution out ofthe way, we will look at some of the main historical threads.

Categories, functors, natural transformations, limits and colimitsappeared almost out of nowhere in a paper by Eilenberg &Mac Lane (1945) entitled “General Theory of NaturalEquivalences.” We say “almost,” because their earlier paper (1942)contains specific functors and natural transformations at work,limited to groups. A desire to clarify and abstract their 1942 resultsled Eilenberg & Mac Lane to devise category theory. Thecentral notion at the time, as their title indicates, was that ofnatural transformation. In order to give a general definition of thelatter, they defined functor, borrowing the term from Carnap, and inorder to define functor, they borrowed the word ‘category’from the philosophy of Aristotle, Kant, and C. S. Peirce, butredefining it mathematically.

After their 1945 paper, it was not clear that the concepts of categorytheory would amount to more than a convenient language; this indeedwas the status quo for about fifteen years. Category theory wasemployed in this manner by Eilenberg & Steenrod (1952), in aninfluential book on the foundations of algebraic topology, and byCartan & Eilenberg (1956), in a ground breaking book onhomological algebra. (Curiously, although Eilenberg & Steenroddefined categories, Cartan & Eilenberg simply assumed them!) Thesebooks allowed new generations of mathematicians to learn algebraictopology and homological algebra directly in the categorical language,and to master the method of diagrams. Indeed, without the method ofdiagram chasing, many results in these two books seem inconceivable,or at the very least would have required a considerably more intricatepresentation.

The situation changed radically with Grothendieck’s (1957) landmarkpaper entitled “Sur quelques points d’algèbre homologique”, inwhich the author employed categories intrinsically to define andconstruct more general theories which he (Grothendieck 1957) thenapplied to specific fields, e.g., to algebraic geometry. Kan (1958)showed that adjoint functors subsume the important concepts of limitsand colimits and could capture fundamental concepts in other areas (inhis case, homotopy theory).

At this point, category theory became more than a convenientlanguage, by virtue of two developments.

1. Employing the axiomatic method and the language of categories,Grothendieck (1957) defined in an abstract fashion types ofcategories, e.g., additive and Abelian categories, showed how toperform various constructions in these categories, and proved variousresults about them. (It should be pointed out that Mac Lane in his1950 paper had made a previous attempt which introduced certain keyideas, for instance using arrows to define certain fundamentalconcepts and that Buchsbaum had essentially introduced the notion of anabelian category independently under the name of “exactcategory” in 1955.) In a nutshell, Grothendieck showed how todevelop part of homological algebra in an abstract setting of thissort. From then on, a specific category of structures, e.g., acategory of sheaves over a topological space \(X\), could be seen as atoken of an abstract category of a certain type, e.g., an Abeliancategory. One could therefore immediately see how the methods of,e.g., homological algebra could be applied to, for instance, algebraicgeometry. Furthermore, it made sense to look for other types ofabstract categories, ones that would encapsulate the fundamental andformal aspects of various mathematical fields in the same way thatAbelian categories encapsulated fundamental aspects of homologicalalgebra.
2. Thanks in large part to the efforts of Freyd and Lawvere, categorytheorists gradually came to see the pervasiveness of the concept ofadjoint functors. Not only does the existence of adjoints to givenfunctors permit definitions of abstract categories (and presumablythose which are defined by such means have a privileged status) but aswe mentioned earlier, many important theorems and even theories invarious fields can be seen as equivalent to the existence of specificfunctors between particular categories. By the early 1970s, theconcept of adjoint functors was seen as central to categorytheory.

With these developments, category theory became an autonomous field ofresearch, and pure category theory could be developed. And indeed, itdid grow rapidly as a discipline, but also in its applications, mainlyin its source contexts, namely algebraic topology and homologicalalgebra, but also in algebraic geometry and, after the appearance ofLawvere’s Ph.D. thesis, in universal algebra. This thesis alsoconstitutes a landmark in this history of the field, for in it Lawvereproposed the category of categories as a foundation for categorytheory, set theory and, thus, the whole of mathematics, as well asusing categories for the study of the logical aspects ofmathematics.

Over the course of the 1960s, Lawvere outlined the basic frameworkfor an entirely original approach to logic and the foundations ofmathematics. He achieved the following:

• Axiomatized the category of sets (Lawvere 1964) and of categories(Lawvere 1966);
• Gave a categorical description of theories that was independent ofsyntactical choices and sketched how completeness theorems for logicalsystems could be obtained by categorical methods (Lawvere 1967);
• Characterized Cartesian closed categories and showed theirconnections to logical systems and various logical paradoxes (Lawvere1969);
• Showed that the quantifiers and the comprehension schemes could becaptured as adjoint functors to given elementary operations (Lawvere1966, 1969, 1970, 1971);
• Argued that adjoint functors should generally play a majorfoundational role through the notion of “categorical doctrines”(Lawvere 1969).

Meanwhile, Lambek (1968, 1969, 1972) described categories in terms ofdeductive systems and employed categorical methods forproof-theoretical purposes.

All this work culminated in another notion, thanks to Grothendieck andhis school: that of atopos. Even though toposes appeared inthe 1960s, in the context of algebraic geometry, again from the mindof Grothendieck, it was certainly Lawvere and Tierney’s (1972)elementary axiomatization of a topos which gave impetus to itsattaining foundational status. Very roughly, an elementary topos is acategory possessing a logical structure sufficiently rich to developmost of “ordinary mathematics”, that is, most of what istaught to mathematics undergraduates. As such, an elementary topos canbe thought of as a categorical theory of sets. But it is also ageneralized topological space, thus providing a direct connectionbetween logic and geometry. (For more on the history of categoricallogic, see Marquis & Reyes 2012, Bell 2005.)

The 1970s saw the development and application of the topos concept inmany different directions. The very first applications outsidealgebraic geometry were in set theory, where various independenceresults were recast in terms of topos (Tierney 1972, Bunge 1974, butalso Blass & Scedrov 1989, Blass & Scedrov 1992, Freyd 1980,Mac Lane & Moerdijk 1992, Scedrov 1984). Connections withintuitionistic and, more generally constructive mathematics were notedearly on, and toposes are still used to investigate models of variousaspects of intuitionism and constructivism (Lambek & Scott 1986,Mac Lane & Moerdijk 1992, Van der Hoeven & Moerdijk1984a, 1984b, 1984c, Moerdijk 1984, Moerdijk 1995a, Moerdijk 1998,Moerdijk & Palmgren 1997, Moerdijk & Palmgren 2002, Palmgren2012, Palmgren 2018. For more on the history of topos theory, see McLarty 1992 and Bell 2012).

More recently, topos theory has been employed to investigate variousforms of constructive mathematics or set theory (Joyal & Moerdijk1995, Taylor 1996, Awodey 2008), recursiveness, and models ofhigher-order type theories generally. The introduction of theso-called “effective topos” and the search for axioms forsynthetic domain theory are worth mentioning (Hyland 1982, Hyland1988, 1991, Hylandet al. 1990, McLarty 1992, Jacobs1999, Van Oosten 2008, Van Oosten 2002 and the referencestherein). Lawvere’s early motivation was to provide a new foundationfor differential geometry, a lively research area which is now called“synthetic differential geometry” (Lawvere 2000, 2002,Kock 2006, Bell 1988, 1995, 1998, 2001, Moerdijk & Reyes1991). This is only the tip of the iceberg; toposes could prove to befor the 21st century what Lie groups were to the 20th century.

From the 1980s to the present, category theory has found newapplications. In theoretical computer science, category theory is nowfirmly rooted, and contributes, among other things, to the developmentof new logical systems and to the semantics of programming. (Pitts2000, Plotkin 2000, Scott 2000, and the references therein). Itsapplications to mathematics are becoming more diverse, even touchingon theoretical physics, which employs higher-dimensional categorytheory — which is to category theory what higher-dimensionalgeometry is to plane geometry — to study the so-called “quantumgroups” and quantum field theory (Majid 1995, Baez & Dolan 2001 and otherpublications by these authors).

3. Philosophical Significance

Category theory challenges philosophers in two ways, which are notnecessarily mutually exclusive. On the one hand, it is certainly thetask of philosophy to clarify the general epistemological andontological status of categories and categorical methods, both in thepractice of mathematics and in the foundational landscape. On theother hand, philosophers and philosophical logicians can employcategory theory and categorical logic to explore philosophical andlogical problems. I now discuss these challenges, briefly, in turn.

Category theory is now a common tool in the mathematician’stoolbox; that much is clear. It is also clear that category theoryorganizes and unifies much of mathematics. Contemporary mathematicalfields would not be what they are without category theory, forinstance algebraic topology, homological algebra, homotopy theory andhomotopical algebra, representation theory, arithmetic geometry andalgebraic geometry. (See for instance Mac Lane 1971, 1998 orPedicchio & Tholen 2004.) No one will deny these simple facts.Furthermore, vast portions of contemporary mathematics now rest on adifferent practice which rely, in large part, on the manipulation ofnew graphical notations, on the one hand, and on different levels ofabstraction, on the other hand. It is not simply that category theoryand the mathematical disciplines developed within that framework usecommutative diagrams, although this in itself leads to someinteresting philosophical explorations, as for instance in De Toffoli2017, but category theorists have seen the need to develop systematicand formal graphical languages to express directly various forms ofargumentations. (See, for instance, Joyal & Street 1993; Joyal,Street & Verity 1996; Fong & Spivak 2019, Other InternetResources.) Whereas sinceBourbaki, mathematics was done “up to isomorphism”, insome cases, it is now done “up to equivalence” or up to“bi-equivalence” or even up to“n-equivalence”. (For an attempt at clarifying what theselevels of abstraction mean, see Marquis 2014, Marquis 2016.)

Doing mathematics in a categorical framework is almost alwaysradically different from doing it in a set-theoretical framework (theexception being working with the internal language of a Boolean topos;whenever the topos is not Boolean, then the main difference lies inthe fact that the logic isintuitionistic). Hence, as isoften the case when a different conceptual framework is adopted, manybasic issues regarding the nature of the objects studied, the natureof the knowledge involved, and the nature of the methods used have tobe reevaluated. We will take up these three aspects in turn.

Two facets of the nature of mathematical objects within a categoricalframework have to be emphasized. First, objects are always given in acategory. An object exists in and depends upon an ambientcategory. Furthermore, an object is characterized by the morphismsgoing in it and/or the morphisms coming out of it. Second, objects arealways characterized up to isomorphism (in the best cases, up to aunique isomorphism). There is no such thing, for instance, asthe natural numbers. However, it can be argued that there issuch a thing asthe concept of natural numbers. Indeed, theconcept of natural numbers can be given unambiguously, via theDedekind-Peano-Lawvere axioms, but what this concept refers to inspecific cases depends on the context in which it is interpreted,e.g., the category of sets or a topos of sheaves over a topologicalspace. Thus, it seems that sense does not determine reference in acategorical context. It is hard to resist the temptation to think thatcategory theory embodies a form of structuralism, that it describesmathematical objects as structures since the latter, presumably, arealways characterized up to isomorphism. Thus, the key here has to dowith the kind of criterion of identity at work within a categoricalframework and how it resembles any criterion given for objects whichare thought of as forms in general. One of the standard objectionspresented against this view is that if objects are thought of asstructures and only asabstract structures, meaning here thatthey are separated from any specific or concrete representation, thenit is impossible to locate them within the mathematical universe. (SeeHellman 2003 for a standard formulation of the objection, McLarty1993, Awodey 2004, Landry & Marquis 2005, Shapiro 2005, Landry2011, Linnebo & Pettigrew 2011, Hellman 2011, Shapiro 2011,McLarty 2011, Logan 2015 for relevant material on the issue.)

A slightly different way to makesense of the situation is to think of mathematical objects astypes for which there are tokens given in differentcontexts. This is strikingly different from the situation one finds inset theory, in which mathematical objects are defined uniquely andtheir reference is given directly. Although one can make room fortypes within set theory via equivalence classes or isomorphism typesin general, thebasic criterion of identity within thatframework is given by the axiom of extensionality and thus,ultimately, reference is made to specific sets. Furthermore, it canbe argued that the relation between a type and its token isnot represented adequately by the membership relation. Atoken does not belong to a type, it is not an element of a type, butrather it is an instance of it. In a categorical framework, one alwaysrefers to atoken of a type, and what the theorycharacterizes directly is the type, not the tokens. In this framework,one does not have to locate a type, but tokens of it are, at least inmathematics, epistemologically required. This is simply the reflectionof the interaction between the abstract and the concrete in theepistemological sense (and not the ontological sense of these latterexpressions.) (See Ellerman 1988, Ellerman 2017, Marquis 2000, Marquis2006, Marquis 2013.)

The history of category theory offers a rich source of information toexplore and take into account for an historically sensitiveepistemology of mathematics. It is hard to imagine, for instance, howalgebraic geometry and algebraic topology could have become what theyare now without categorical tools. (See, for instance, Carter 2008,Corfield 2003, Krömer 2007, Marquis 2009, McLarty 1994, McLarty2006.) Category theory has lead to reconceptualizations of variousareas of mathematics based on purely abstract foundations. Moreover,when developed in a categorical framework, traditional boundariesbetween disciplines are shattered and reconfigured; to mention but oneimportant example, topos theory provides a direct bridge betweenalgebraic geometry and logic, to the point where certain results inalgebraic geometry are directly translated into logic and vice versa.Certain concepts that were geometrical in origin are more clearly seenas logical (for example, the notion of coherent topos). Algebraictopology also lurks in the background. (See, for instance, Caramello2018 for a systematic exploitation of the idea of toposes as bridgesin mathematics.) On a different but important front, it can be arguedthat the distinction between mathematics and metamathematics cannot bearticulated in the way it has been. All these issues have to bereconsidered and reevaluated.

Moving closer to mathematical practice, category theory allowed forthe development of methods that have changed and continue to changethe face of mathematics. It could be argued that category theoryrepresents the culmination of one of deepest and most powerfultendencies in twentieth century mathematical thought: the search forthe most general and abstract ingredients in a givensituation. Category theory is, in this sense, the legitimate heir ofthe Dedekind-Hilbert-Noether-Bourbaki tradition, with its emphasis onthe axiomatic method and algebraic structures. (For a differentreading, see Rodin 2014.) When used to characterize a specificmathematical domain, category theory reveals the frame upon which thatarea is built, the overall structure presiding to its stability,strength and coherence. The structure of this specific area, in asense, might not need to rest on anything, that is, on some solidsoil, for it might very well be just one part of a larger network thatis without any Archimedean point, as if floating in space. To use awell-known metaphor: from a categorical point of view, Neurath’sship has become a spaceship.

Still, it remains to be seen whether category theory should be “on thesame plane,” so to speak, as set theory, whether it should be taken asa serious alternative to set theory as a foundation for mathematics,or whether it is foundational in a different sense altogether. (Thatthis very question applies even more forcefully to topos theory willnot detain us.)

Lawvere from early on promoted the idea that a category of categoriescould be used as a foundational framework. (See Lawvere 1964, 1966.)This proposal now rests in part on the development ofhigher-dimensional categories, also called weak \(n\)-categories.(See, for instance Makkai 1998.) The advent of topos theory in theseventies brought new possibilities. Mac Lane has suggestedthat certain toposes be considered as a genuine foundation formathematics. (See Mac Lane 1986.) Lambek proposed the so-calledfree topos as the best possible framework, in the sense thatmathematicians with different philosophical outlooks might nonethelessagree to adopt it. (See Couture & Lambek 1991, 1992, Lambek1994.) He has also argued that there is no topos that canthoroughly satisfy a classical mathematician. (See Lambek 2004.) (Formore on the various foundational views among category theorists, seeLandry & Marquis 2005.)

Arguments have been advanced for and against category theory as afoundational framework. (Blass 1984 surveys the relationships betweencategory theory and set theory. Feferman 1977, Bell 1981, and Hellman2003 argue against category theory. See Marquis 1995 for a quickoverview and proposal and McLarty 2004 and Awodey 2004 for replies toHellman 2003.) The debate has advanced slowly but surely. It has beenrecognized that it is possible to present a foundational framework inthe language of category theory, be it in the form of the ElementaryTheory of the Category of Sets, ETCS, or a category of categories, ofMakkai Structuralist foundations for abstract mathematics, SFAM. Thus,it seems that the community no longer question the logical and theconceptual autonomy of these approaches, to use the terminologyintroduced by Linnebo & Pettigrew 2011. The main issue seems to bewhether one can provide a philosophically satisfying justification forone of those framework. (See Hellman 2013, Landry 2013, Marquis 2013b,McLarty 2018.)

This matter is further complicated by the fact that the foundations ofcategory theory itself have yet to be clarified. For there may be manydifferent ways to think of a universe of higher-dimensional categoriesas a foundations for mathematics. It is safe to say that we now have agood understanding of what are called \((\infty, 1)\)-categories and importantmathematical results have been obtained in that framework. (See, forinstance, Cisinski 2019 for a presentation.) An adequate language forthe universe of arbitrary higher-dimensional categories still has tobe presented together with definite axioms for mathematics. (SeeMakkai 1998 for a short description of such a language. A differentapproach based on homotopy theory but with closed connections withhigher-dimensional categories has been proposed by Voevodsky etal. and is being vigorously pursued. See the bookHomotopy TypeTheory, by Awodey et al. 2013.)

It is an established fact that category theory is employed to studylogic and philosophy. Indeed, categorical logic, the study of logic bycategorical means, has been under way for about 40 years now and stillvigorous. Some of the philosophically relevant results obtained incategorical logic are:

• The hierarchy of categorical doctrines: regular categories,coherent categories, Heyting categories and Boolean categories; allthese correspond to well-defined logical systems, together withdeductive systems and completeness theorems; they suggest that logicalnotions, including quantifiers, arise naturally in a specific orderand are not haphazardly organized (see Walsh 2017 for a philosophicaljustification of logical connectives using category theory andHalvorson & Tsementzis 2018 for a look from the point of view ofscientific theories);
• Joyal’s generalization of Kripke-Beth semantics for intuitionisticlogic to sheaf semantics (Lambek & Scott 1986, Mac Lane& Moerdijk 1992);
• Coherent and geometric logic, so-called, whose practical andconceptual significance has yet to be explored (Makkai & Reyes1977, Mac Lane & Moerdiejk 1992, Johnstone 2002, Caramello2011b, 2012a);
• The notions of generic model and classifying topos of a theory(Makkai & Reyes 1977, Boileau & Joyal 1981, Bell 1988, Mac Lane& Moerdijk 1992, Johnstone 2002, Caramello 2012b);
• The notion of strong conceptual completeness and the associatedtheorems (Makkai & Reyes 1977, Butz & Moerdijk 1999, Makkai1981, Pitts 1989, Johnstone 2002);
• Geometric proofs of the independence of the continuum hypothesisand other strong axioms of set theory (Tierney 1972, Bunge 1974, Freyd1980, 1987, Blass & Scedrov 1983, 1989, 1992, Mac Lane &Moerdijk 1992);
• Models and development of constructive mathematics (seebibliography below);
• Synthetic differential geometry, an alternative to standard andnon-standard analysis (Kock 1981, Bell 1998, 2001, 2006);
• The construction of the so-called effective topos, in which everyfunction on the natural numbers is recursive (McLarty 1992,Hyland 1982, 1991, Van Oosten 2002, Van Oosten 2008);
• Categorical models of linear logic, modal logic, fuzzy sets, andgeneral higher-order type theories (Reyes 1991, Reyes & Zawadoski1993, Reyes & Zolfaghari 1991, 1996, Makkai & Reyes 1995,Ghilardi & Zawadowski 2002, Rodabaugh & Klement 2003, Jacobs1999, Taylor 1999, Johnstone 2002, Blute & Scott 2004, Awodey& Warren 2009, Awodey et al. 2013, Kishida 2018, Cockett &Seely 2018);
• A graphical syntax called “sketches” (Barr & Wells1985, 1999, Makkai 1997a, 1997b, 1997c, Johnstone 2002).
• Quantum logic, the foundations of quantum physics and quantumfield theory (Brunetti et al. 2003, Abramsky & Duncan 2006, Heunen et al. 2009, Baez& Stay 2010, Baez & Lauda 2011, Coecke 2011, Isham 2011,Döring 2011, Eva 2017, Coecke & Kissinger 2018).

Categorical tools in logic offer considerable flexibility, as isillustrated by the fact that almost all the surprising results ofconstructive and intuitionistic mathematics can be modeled in a propercategorical setting. At the same time, the standard set-theoreticnotions, e.g. Tarski’s semantics, have found natural generalizationsin categories. Thus, categorical logic has roots in logic as it wasdeveloped in the twentieth century, while at the same time providing apowerful and novel framework with numerous links to other parts ofmathematics.

Category theory also bears on more general philosophical questions.From the foregoing disussion, it should be obvious that categorytheory and categorical logic ought to have an impact on almost allissues arising in philosophy of logic: from the nature of identitycriteria to the question of alternative logics, category theory alwayssheds a new light on these topics. Similar remarks can be made when weturn to ontology, in particular formal ontology: the part/wholerelation, boundaries of systems, ideas of space, etc. Ellerman (1988)has bravely attempted to show that category theory constitutes atheory of universals, one having properties radically different fromset theory, which is also seen as a theory of universals. Moving fromontology to cognitive science, MacNamara & Reyes (1994) have triedto employ categorical logic to provide a different logic ofreference. In particular, they have attempted to clarify therelationships between count nouns and mass terms. Other researchersare using category theory to study complex systems, cognitive neuralnetworks, and analogies. (See, for instance, Ehresmann 2018, Ehresmann &Vanbremeersch 1987, 2007, Healy 2000, Healy & Caudell 2006,Arzi-Gonczarowski 1999, Brown & Porter 2006.) Finally,philosophers of science have turned to category theory to shed a newlight on issues related to structuralism in science. (See, forinstance, Brading & Landry 2006, Bain 2013, Lam & Wüthrich2015, Eva 2016, Lal & Teh 2017, Landry 2007, 2012, 2018.)

Category theory offers thus many philosophical challenges, challengeswhich will hopefully be taken up in years to come.

Bibliography

Readers may find the following useful:

Programmatic Reading Guide

The citations in this guide and in the text above can all befound in the list below.

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• Cheng, E. & Lauda, A., 2004,Higher-Dimensional Categories: an illustrated guide book.
• Fong, B. and Spivak, D., 2019, “Graphical RegularLogic”, available at arXiv.org.
• The category theory mailing list (with many links and useful information)
• The n-category Cafe, A blog on higher-dimensional categories, physics and philosophy
• The nLab (an open access resource on categories and higher-dimensional categories)
• Logic Matters Peter Smith’s blog on logic. It contains a page on category theory with many references and discussions.
• The Catsters’ Category Theory Videos, introducing the basic concepts of category theory
• Math3Ma A math blog that contains introductions to the main concepts of category theory, set theory and other mathematical topics

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