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Appendix. Foundations We have described a category in terms of sets, as a set of objects and a set of arrows. However, categories can be described directly-and they can then be used as a possible foundation for all of mathematics, thus replacing the use in such a foundation of the usual Zermelo-Fraenkel axioms for set theory. Here is the direct description: 1. Objects and arrows. A category consists of objects a, b, c, ... and arrows f, g, h. Sets form a category with sets as the objects and functions as the arrows. 2. Domain. Each arrow f has an object a as its "domain" and an object b as its "codomain"; we then write f : a --+ b. 3. Composition. Given f : a --+ band g : b --+ c, their composite is an arrow g 0 f : a --+ c. 4. Associativity. If also h: c --+ d, then the triple composition is associative: h 0 (g 0 f) = (h 0 g) 0 f : a --+ d . 5. Identities. Each object b has an identity arrow Ib: b --+ b. If also f : a --+ b, then Ib 0 f = f. If also g : b --+ c, then go Ib = g. An elementary topos is a category with a certain additional structure: terminal object, pullbacks, truth, a subobject classifier, and power objects (sets of subsets). The axioms for this addtional structure are as follows: 6. Terminal object. There is a terminal object 1 such that every ob­ ject a has exactly one arrow a --+ 1. 7. Pullbacks. Every pair of arrows f: a --+ b +- c : g with a common codomain b has a pullback, defined as in § 111.4: (1) a ----+ b. f In particular, (take b = 1), any two objects a and c have a product a x c. 8. Truth. There is an object Q (the object of truth values) and a 289 290 Appendix. Foundations monomorphism t: 1 -+ Q called truth; to any monomorphism m : a -+ b, there is a unique arrow ljJ : b -+ Q such that the fol­ lowing square is a pullback: a -----+ 1 mi It (2) b -----+ Q. '" 9. Power objects. To each object b, there is an associated object P b and an arrow eb : b x P b -+ Q such that for every arrow f : b x a -+ Q there is a unique arrow g : a -+ P b for which the following diagram commutes: f bxa -----+ Q (3) lxgI II b x Pb -----+ Q. tb To understand these axioms, we observe how they apply to the usual category of all sets. There, any set with just one element can serve as a terminal object 1, because each set a has a unique function a -+ 1 to 1. For two sets a and b, the pullback of two arrows a -+ 1 +-- b is then the usual set-theoretic product, with its projections to the given factors a and b. For truth values, take the object Q to be any set 2 consisting of two objects, 1 and 0, while the monomorphism t : 1 -+ 2 is just the usual in­ clusion of 1 in 2. Then a monomorphism m : a -+ b, as in Axiom 8, is a subset a of b. This subset has a well-known characteristic function ljJ : b -+ 2 with ljJ(y) = 1 or 0 according as the element of y of b is or is not in the subset a. This produces the pullback (2) above. Axiom 9 describes P b, the set of all subsets s of b, often called the "power set" Pb. Indeed, one can then set eb(x, s) = 0 if the element x of b is in the subset s and equal to 1 otherwise. This does give a pullback, as in (3) above. These axioms for a topos then hold for the category of sets. They have a number of strong consequences. For example, they give all finite cate­ gorical products and pullbacks, as well as all finite coproducts, including an initial object 0, the empty set. For example, they provide a right adjoint to the product a x b as a functor of a; this is the exponential ch with (see § IV.6) hom(a x b, c) ~ (a, cb ) . A category of sets can now be described as an elementary topos, defined as above, with three additional properties: Appendix. Foundations 291 (a) it is well-pointed, (b) it has the axiom of choice (AC), (c) it has a natural-numbers object (NNO). In describing these properties, it is useful to think of the objects as sets and the arrows as functions. Well-pointed requires that if two arrows f, g : a -+ b have (f"# g) then there must exist an arrow p : 1 -+ a for whichfp "# gpo The intention is that when the functions f and g differ they must differ at some "point" p, that is, at some element p of the set a. The axiom of choice (AC) requires that every subjectionf : a -+ b has a right inverse 'l" : b -+ a for which f 0 'l" = b. This right inverse picks out to each point p : 1 -+ b of b a point of a, to wit, the composite 'l"P which is mapped by f onto p. The natural-numbers object (NNO) can be described as a set N with an initial object 0 : 1 -+ N and a successor function s : N -+ N in terms of which functions f : N -+ X on N can be defined by recursion, by specifying f(O) and the composite fos. In other words, an NNO N in a category is a diagram 1 !!..,N':"'N consisting of a point 0 of N and a map s such that, given any arrows 1 !!... b .!:.. b, there is a unique arrow f : N -+ b which makes the following diagram commute: 1~ II 1~ In the usual functional notation, this states that fO=h, fs=kf; that is,fis defined by givingf( 0) and thenf(n + 1) in terms off(n). Thus, the category of sets may be described as a well-pointed topos with the AC in which there is an NNO. This set of axioms for set theory is weakly consistent with a version of the Zermelo axioms (the so-called "bounded" Zermelo; see Mac Lane and Moerdijk [1992]). They are originally due to Lawvere [1964], who called them the "elementary theory of the category of sets." Among the other examples of an elementary topos are the sheaves on a topological space; see Mac Lane and Moerdijk [1992]. In these axioms, it is often assumed that the subject classifier Q has just two elements. This makes it a Boolean algebra. Table of Standard Categories: Objects and Arrows Ab Abelian groups Adj Small categories, adjunctions, p. 104 Alg <, E)-algebras Bool Boolean algebras CAb Compact topological abelian groups CAT Categories, functors CG Haus Compactly generated Hausdorff spaces p. 185 Comp Bool Complete Boolean algebras Comp Haus Compact Hausdorff spaces, p. 125 CRng Commutatative rings, homomorphisms Ensv Sets and functions, within a universe V, p. 11 Euclid Euclidean vector spaces, orthogonal transformations Fin Skeletal category of finite sets Finord Finite ordinals, all set functions, p. 12 Grp Groups and homomorphisms Grpb Directed graphs and morphisms Haus Hausdorff Spaces, continuous maps Lconn Locally connected topological spaces K-Mod K-Modules and their morphisms Mod-R Right R-modules, R a ring R-Mod Left R-modules and morphisms MatrK Natural numbers, morphisms rectangular matrices, p. 11 Mon Monoids and morphisms of monoids, p. 12 Moncat Monoidal categories and strict morphisms, p. 160 Ord Ordered sets, order-preserving maps, p. 123 Rng Rings and homomorphisms Ses-A Short exact sequences of A -modules Set All small sets and functions Set. Sets with base point Smgrp Semigroups and morphisms Top Topological spaces, continuous maps, p. 122 Topb Topological spaces, homotopy classes of maps, p. 12 Vet Vector spaces, linear transformations o Empty category, p. 10 1 One-object category, p. 10 2 Two objects, p. 10 3 Three objects, p. 11 <Q,E) Universal algebras, type r, p. 120, 152 b 1 c Objects of C under b, p. 46 c 1 a Objects of C over a, p. 46 T1S Comma category, p. 46 293 Table of Terminology This Book Elsewhere (for abbreviations, see below) arrow map (E & M), morphism (Gr) domain source (Ehr) codomain target (Ehr) graph precategory, diagram scheme (Mit) natural transformation morphism of functors (Gr), functorial map (G-Z) natural isomorphism natural equivalence (E & M; now obsolete) mOlllc monomorphism epi epimorphism, epic idempotent projector (Gr) opposite dual coproduct sum equalizer kernel, difference kernel pullback fibered product (Gr), cartesian square pushout cocartesian square, comeet universal arrow left liberty map (G-Z) limit exists limit is representable (Gr) limit projective limit, inverse limit colimit inductive limit, direct limit cone to a functor projective cone, inverse cone (G-Z) cone from a functor inductive cone, co-cone left adjoint coadjoint (Mit), adjoint right adjoint adjoint (Mit), coadjoint unit of adjunction adjunction morphism (G-Z) triangular identities 8 quasi-inverse to IJ (G-Z) monad triple biproduct direct sum (in Ab-categories) Ab-category preadditive category (old) Gr = Grothendieck Ehr = Ehresmann Mit = Mitchell E & M = Eilenberg & Mac Lane G-Z = Gabriel-Zisman 295 Bibliography Adamek, J., Herrlich, H., Strecker, G.B. [1990]: Abstract and concrete cate­ gories. New York: John Wiley & Sons 1990. Andre, Michel [1967]: Methode simpliciale en algebre homologique et algebre commutative IV. Lecture Notes in Mathematics, Vol. 32. Berlin-Heidelberg­ New York: Springer 1967. - [1970]: Homology of simplicial objects. Proceeding of the AMS symposium in Pure Mathematics on Applications of Categorical Algebra, pp. 15-36.
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