FORMAL CATEGORY THEORY

Transcription

1 FORML TEGORY THEORY OURSE HELD T MSRYK UNIVERSITY IVN DI LIERTI, SIMON HENRY, MIKE LIEERMNN, FOSO LOREGIN bstract. These are the notes o a readin seminar on ormal cateory theory we are runnin at Masaryk University. It is not live-texed, and yet it is ull o misprints, reerences to nameless cults, blood stains, blasphemy. 1. Ouverture: what is ormal cateory theory The lanuae o cateory theory is built upon a certain number o undamental notions: amon these we ind the universal characterization o co/limits, the deinition o adjunction, (pointwise) Kan extension, and the theory o monads. It is possible to axiomatize these deinitions, pretendin that they reer to the 1- and 2-cells o a eneric 2-cateory other than at. This conceptualization is one o the pillars upon which cateory theory is done: in some sense, cateory theory arises when the way in which abstract patterns interact becomes itsel an object o study, and when it is eneralized to several dierent contexts. In a ew words, the aim o ormal cateory theory is to provide a ramework in which this process can be outlined mathematically. Quotin the introduction o [Gra74], that is one o the pillars on which the subject is ounded, The purpose o cateory theory is to try to describe certain eneral aspects o the structure o mathematics. Since cateory theory is also part o mathematics, this cateorical type o description should apply to it as well as to other parts o mathematics. [...] The basic idea is that the cateory o small cateories, at, is a 2-cateory with properties in the same way Set is a cateory with properties. The aim o ormal cateory theory is to outline these properties, and the assumptions needed to ensure that a certain 2-cateory behaves like at in some or some other respects. Unortunately, bein too naïve when perormin this process doesn t always ive the riht answer (by which we mean that it doesn t always build an object with the riht universal property), or at least it doesn t ive the riht answer in the same straihtorward way in which some cateories o alebraic structures can be deined startin rom the cateory o Set. This is ultimately due to the act that, when movin to the settin o V-enriched cateories (which is the adjacent step o abstraction rom at = Set-at) the theory behaves dierently in various ways, and some o these dierences prevent V-cateories to be as expressive as one would have liked it to be (a paradimatic example o this minor expressivity is the lack o a Grothendieck construction or eneric V-presheaves: seein how the Grothendieck construction, a certain rule to relate presheaves on and ibrations over, ultimately pertains to ormal cateory theory has been one o the purposes o the early literature on the subject, see [Str74, SW78, Str80]). The major problem is that the 2-cateory V-at oten doesn t ive enouh inormation about the V-valued hom-unctors in a 2-cateory. Formal cateory Date: Wednesday 18 th October,

2 2 IVN DI LIERTI, SIMON HENRY, MIKE LIEERMNN, FOSO LOREGIN theory can be thouht as a way to encode the same amount o inormation in various other ways: even thouh it is always possible to do some constructions by mimickin deinitions rom at (adjunctions and adjoint equivalences, extensions by universal 2-cells, etc.), thins et a little hairy when we want to provide the theory with an analoue o a very useul and basic result as the Yoneda lemma. In the 2-cateory at, we can use a lax limit construction to revert set-valued unctors on an object into arrows over (we basically lue toether a bunch o ibers b E b projectin onto, in the same manner we build the étale space o a preshea F : op Set); in the 2-cateory at the comma object o b : 1 to 1 : toether with its projection b/ is a ood stand in or the covariant unctor represented by (more enerally, discrete let ibrations over stand in or eneral unctors Set). In the 2-cateory V-at, we care about V-valued V-unctors and we would like to do the same construction there. ut or an object b in a V-enriched cateory, the comma b/ is more naturally an internal cateory (whose object o objects is x (b, x)) rather than an enriched one (whose objects are morphisms b x in the underlyin cateory o ). We have to ensure that the codomain projection V-unctor b/ rom the enriched version o the comma has a ibration-like properties, and this leaves us with the undamental problem o ormal cateory theory: which additional structure on a 2-cateory K allows to reconize arrows o K playin the same rôle o discrete ibrations in at, thus providin with a meaninul notion o Yoneda lemma internal to K? It has been in the middle ae o southern-emisphere cateory theory that a certain number o ways to describe such extra structures have been invented: the aim o this irst chapters is to ive a brie account about three (not unrelated) such attempts. t the moment o writin we count 1.1. Street and Walters Yoneda structures. The deinition o Yoneda structure iven by Street revolves around the possibility to ive a ormal counterpart o the Yoneda embeddin よ : P with its universal property. This axiomatization is based on the centrality o the Yoneda lemma internal to a 2-cateory K, that has been deined in a airly heavy-handed way in Street s previous [Str74]. One o the main achievements o the subsequent [SW78] is to obtain an eleant and concise axiomatization stemmin almost completely rom universal properties Street s ibrational cosmoi. particular case o Yoneda structures, where you ask the pseudo-unctor P : K coop K characterizin a Yoneda structure to be a riht 2-adjoint Wood s proarrow equipments. dierent and more powerul perspective, where you embed K into a second 2-cateory K with a 2-unctor ( ) : K K, which is the identity on objects and mimicks the behaviour o the embeddin at Pro, askin that the 2-unctor ( ) is locally ully aithul; or each 1-cell : in K, every admits a riht adjoint in K. (See [éner, Lor15] or a thorouh account o the theory o prounctors; since the request that ( ) is the identity on objects is a bit o an evil one, the paper [Woo85] ives the 2-unctors satisyin only the other two axioms the name o pro-equipments). 2. Yoneda structures on 2-cateories 2.1. Lit, extension, contraction, expansion.

3 FORML TEGORY THEORY 3 Deinition 2.1 : Let a cospan o 1-cells in K. let litin o alon consists o a pair lit, (oten denoted simply as lit ) initial amon the commutative trianles like the one below: lit lit h h In other words, composition with : lit determines a bijection γ ( γ) between 2-cells lit γ h and 2-cells h. Remark 2.2 : One can deine riht litins similarly, reversin only the direction o the 2-cell in the diaram above, and consequently the universal property, and let and riht extensions reversin, respectively, only the directions o 1-cells or the direction o both 1- and 2-cells in the diaram above. It is then clear that let extensions in K are let litins in K op, riht litins in K are let litins in K co, and riht extensions are let litins in K coop. The situation is conveniently depicted in the ollowin array o universal objects: ε Lan Ran Lan h h h h Ran Lit h h h Rit H Lit Rit ε Deinition 2.3 : There is an obvious notion o preservation o a let litin lit (write it down abstractin a little bit rom the deinition o preservation o a co/limit) under the composition with a 1-cell u; we say that a let litin is absolute i it is preserved by every u. O course similar deinitions apply to riht litins and let or riht extensions Three standard results on lits and extensions. Recall that we call : X Y : a pair o adjoint 1-cells i we are iven a 2-cell ɛ : 1 and : 1 satisyin the zi-za identities ε = 1 and ε = 1. We denote this situation in the compact orm ε : X Y. Lemma 2.4 [The most intrinsic characterization o adjointness you could ever think o]: The ollowin conditions are equivalent: (1) is the let litin o the identity 1 alon : and this litin is preserved by :. (2) is the absolute let litin o the identity 1 alon : ; (3) ε : ; Proo. It is obvious that (2) (1). We then prove that (1) (3) and (3) (2). The trick is to ind ɛ : 1 satisyin the zi-za identities: since, is the let litin lit, there is a unique such ɛ such that the equation ɛ = 1 holds in the diaram below:

4 4 IVN DI LIERTI, SIMON HENRY, MIKE LIEERMNN, FOSO LOREGIN 1 ɛ This is one o the zi-za identities testiyin that ɛ identity can be obtained rom the chain o equalities. The other zi-za 1 a ε 1 a ɛ 1 a but now ε = 1 a by uniqueness ( induces a bijection Nat(, ) = Nat(1, )). This shows that ɛ. ssumin that ɛ, we must show that, is an absolute extension. It is an extension, since iven a unctor h : X, and endin the bijections (a, ha) = (a, ha) we et that Nat(1, h) = Nat(, h); written explicitly, the bijection sends α : 1 h into its mate α = ε h F α : h h, and the uniqueness is iven by the bijectivity o α α. similar arument shows that this litin is absolute, as Nat(h, k) = (hx, kx) x = (hx, kx) x = Nat(h, k); this shows that h, h is lit h, h. Lemma 2.5 [ pastin lemma or let extensions]: Given the diaram o natural transormations between unctors h D k assume that the external trianle, plus the let trianle are let extensions, i.e. there are 2-cells : h k such that k = Lan h and β : h n such that n = Lan h. Then the riht trianle is a let extension, meanin that There is a unique ˆβ : k n such that β = ˆβ ; Such ˆβ makes the pair (n, ˆβ) a let extension o k alon. Proo. Exercise. Use the universal properties you already have to supply the additional one. (dditional question: is it still true or absolute extensions?) Deinition 2.6 [relative adjunction]: Let : X Y, j : X, and : Y be three unctors; we say that is a j-relative let adjoint to, and we write [j], i there is a natural isomorphism Y (x, y) = X(jx, y). β n

5 FORML TEGORY THEORY 5 Remark 2.7 : It is a matter o checkin universal properties to prove that [j] i exhibits rit j. This ives a ormal characterization o relative adjoints eneralizin Lemma 2.4. It is o course possible to deine a relative riht adjoint: iven X Y and j : Y, we write that [j]ε i Y (x, jc) = X(x, c). Then, [j]ε i and only i = lit j Remark 2.8 : The deinition o relative adjunction is eminently asymmetric: the most important dierence is that the two unctors do not determine each other any more, as it is only true that i [j]ε then determines uniquely, and not viceversa (immediate in view o the ormal characterization above). part rom this, and keepin in mind this inevitable asymmetry, pretty much all the theory o adjoint unctors can be relativized: Relative adjunctions have units or counits, dependin on whether they are let or riht; relative adjunctions enerate relative monads in the sense o []; i [j], and j j r then jr [j rj] j r, with a pastin o, Yoneda structures: presentin the axioms. The idea in this section is to present the bare axioms and then show why these are sensible abstractions o trivially true properties o the 2-cateory at. The analoy here is with the motivation or Giraud axioms characterizin a Grothendieck topos, or with the cateorical properties o Set that characterize (a weak version o) it in ETS. Then, we discuss the consequences o the axiom we sinle out showin the most we can in a purely ormal way. We establish the ollowin notation: K is a 2-cateory, ixed once and or all; dm(, ) K (, ) is a ull subcateory o admissible 1-cells, which is moreover a riht ideal, meanin that the composition map restricted to admissible 1-cells ives dm(, ) K (X, ) dm(x, ). We call admissible an object such that 1 dm(, ); notice that this entails that every 1-cell with admissible codomain is itsel admissible. we assume that the ollowin structure can be ound on K : (1) or each admissible object K we can ind an admissible 1-cell よ : P called a Yoneda arrow; (2) or each : admissible 1-cell with admissible domain, we can ind a 2-cell よ χ P (,1) xiom 1 The pair (, 1), χ exhibits lan よ.

6 6 IVN DI LIERTI, SIMON HENRY, MIKE LIEERMNN, FOSO LOREGIN The validity o this axiom in at justiies the notation: indeed, in at the unctor (, 1) amounts precisely to the unctor λb.λa.( a, b). O course, a unctor is admissible i it has small domain, and P is the cateory [ op, Set] o presheaves on. The proo that (, 1) = lan よ, here and elsewhere, will be the result o a nity coend-julin: we have that lan よ (b) = a (a, b) よ (a) = a (F a, b) (, a) = (, b). xiom 1 entails that the correspondence (, 1) is, in a suitable sense, unctorial, as a map dm(, ) op K (X, ) K (X, P) Indeed, iven a 2-cell α : between admissible 1-cells in K, there is a unique ᾱ : (, 1) (, 1) such that the diaram よ χ (,1) c P (,1) = α よ χ P (,1) commutes or a sinle 2-cell induced by the universal property o lan よ, and such 2-cell can quite rihtly be called (α, 1) (notice that the pastin (, 1) α χ o the riht square exists only i α :, so λα.(α, 1) must be contravariant whatever its deinition). Instead, iven a 2-cell X b b β, we deine X (,b) (α,β) P = X (,b ) b b β (,1) (α,1)p (,1) xiom 2 The pair, χ exhibits lit (,1) よ. The validity o this axiom in at is aain a ame o coend calculus: i we call N = lan よ = (, 1) or short, we have lit N N,, where N, : N is the direct imae unctor; then we have Nat ( よ, N ) = [ op, Set] ( よ a, N (a ) ) a = [ op, Set] ( よ a, (, a ) ) a = (a, a ) a = Nat(, )

7 FORML TEGORY THEORY 7 xiom 3 Given a pair o composable 1-cells, the pastin o 2-cells よ P よ χ よ P P χ (,1) exhibits lan よ = (, 1), and the pair 1 P, 1 よ exhibits lan よ よ. The hidden meanin o this axiom is that P is a pseudounctor K coop K. Let s make this evident: iven a pair o composable the universal property o χ entails that there is a unique 2-cell θ illin the diaram よ χ P P θ P P よ xiom 3 is equivalent to the request that this arrow is invertible (exercise: draw the riht diaram), and this yields that the above diaram has the same universal property o よ P よ P P P() which in turn entails that there is a unique, and invertible, 2-cell P() P P. This is o course the irst part o the structure o pseudounctor on P; the remainin structure is iven by the request that 1 P, 1 よ exhibits lan よ よ. Remark 2.9 : s this miht appear quite enimatic, let s recall that we call dense a 1-cell k with the property that lan k k = 1; this allows us to rephrase the second part o axiom 3 sayin that the Yoneda embeddin is dense. This is in act a characterizin property, as the Yoneda lemma is essentially a statement about the inclusion P bein able to enerate all P under colimits. s the universal property o P(1 ), deined above, entails that there is a unique 2-cell ι : P(1 ) 1 P, axiom 3 is equivalent to the request that ι is invertible (and natural in ). This renders P a pseudo-unctor, as claimed above. ll these remarks are o course trivial in at, since the unctoriality o the correspondence  can be proved directly. Nevertheless, axiom 3 is still tellin us somethin about a reduction rule or composition o Kan extensions: indeed, it is possible to prove that (in the same notation o axiom 3) θ : lan よ = lan よ よ lan よ

8 8 IVN DI LIERTI, SIMON HENRY, MIKE LIEERMNN, FOSO LOREGIN There is an additional axiom: xiom 4 Let (,1) σ P be a 2-cell; i it has the property that the pastin よ P exhibits lit よ, then σ is invertible. σ It must be noted that this axiom is mainly useul to make some statements and proos look better: there are reasons not to include it in the deinition o a bare Yoneda structure; it can be proved that 1, 2, 4 3 so that we can call nice Yoneda structures those that satisy 1, 2, and 4. We now concentrate on a ew examples that make evident how cateory theory can be re-enacted in a 2-cateory with a Yoneda structure. In a nice Yoneda structure, we have a nicer characterization o adjoints and a more intuitive analoue o ully aithul 1-cells: I axiom 4 holds, then we recover the characterization o relative adjoints below (see 2.10; in eneral, only one implication holds) in terms o let litins: iven 1-cells :, :, j : we have = lit j i and only i [j], i.e. i and only i (, 1) = (j, ). i axiom 4 holds, then a 1-cell is ully aithul i and only i the unctor K (X, ) is ully aithul or each X, naturally in X Yoneda structures: theorems. reat deal o cateory theory can be developed in a cateory K endowed with a Yoneda structure. We collect here a ew results comin rom [SW78]. Theorem 2.10 [on relative adjoints]: Suppose j :, :, : are 1-cells in K ormin a relative adjunction [j], and,, j are admissible. Then the equality o 2-cells χ (,1) π (j,1) よ P = j χ j よ (j,1) P holds since the let 2-cell deines a よ (j, ), that (by the universal property o χ ) must be o the orm (π ) χ or a unique π : (, 1) (j, ). This determines a bijection π : (, 1) (j, t) : j I π is invertible then the correspondin exhibits lit j. Theorem 2.11 [on adjoints]: Take : : such that, are admissible. Given a 2-cell : 1, the universal property o よ induces a bijection

9 FORML TEGORY THEORY 9 between these s and 2-cells π : (, 1) (1, u): よ よ P (1,u) = よ (,1) π P (1,u) Then is a unit o an adjunction i and only i π is invertible. Moreover, i then or any X K, and a : X, b : X 1-cells, wit X, a, a admissible we have the pointset characterization o adjoints (a, b) = (a, b). Deinition 2.12 [weihted colimit in ormal cateory theory]: Given admissible, : and M, j : M P we deine a j-indexed colimit (or j-weihted colimit) or, and write j : M or a j-relative let adjoint o (, 1). This means that we can write a tensor-hom -like adjunction (j, 1) = P(j, (, 1)) Remark 2.13 : Notice that Thm above characterizes j as an absolute let litin o j alon (, 1); the converse is in eneral not true (it requires axiom 4), and then this let extension deserves the name o weak j-indexed colimit. In view o the above remark, it is obvious what does it mean or a 1-cell h to preserve a j-indexed colimit j. We have the ollowin Theorem 2.14 : let adjoint 1-cell l : preserves all (weak) j-indexed colimits that exist in K and can be composed with l. Proo. The proo is notationally tautoloical (and shows the power o endowin a 2-cateory with a Yoneda structure) in view o the deinition or the let extension X(u, 1) and the composition X(1, u) = よ X u: assume l r is an adjunction with l : X, and that the diaram exhibitin j is iven; then X j j P (,1) P(j, X(l, 1)) = P(j, (, r)) = P(j, (, 1)) r = (j, 1) r = (j, r) = X(l(j ), 1) thus exhibitin j l. Theorem 2.15 : Suppose M,, j : M P, : are admissible. I j exists and i additional admissible N, i : N PM are such that i j exists, then there is an associativity isomorphism i (j ) = (i j). The ollowin theorem is what can be called ninja Yoneda lemma : it amounts to the statement that tensorin with a representable acts like an evaluation.

10 10 IVN DI LIERTI, SIMON HENRY, MIKE LIEERMNN, FOSO LOREGIN Theorem 2.16 [the ninja yoneda lemma]: For admissible, : and X, a : X there is an isomorphism (1, a) = a 2.4. Yoneda structures: examples. Here we collect a ew examples o Yoneda structures or dierent choices o K : (1) the 2-cateory at has a Yoneda structure where P = [ op, Set]. (2) the 2-cateory V-at o cateories enriched over a base V has a Yoneda structure where P = [ op, V] (you basically pretend your proo lives in at = Set-at). (3) the 2-cateory at(k ) o internal cateories in a initely complete K (like or example a topos) has a Yoneda structure whose existence we sketch in Exercise (4) I K has a Yoneda structure and is a small 2-cateory, Psd[ op, K ] has an objectwise Yoneda structure Selected exercises. This subsection collects a ew exercises; the ones labelled with a symbol are to be done with patience, a comortable spot in the library and a cup o ood coee (warnin: merican coee miht be an insuicient adjuvant); the ones labelled with a daner symbol are meant to be borin technicalities no one ever checks, or extremely diicult exercises. Exercise 2.17 : Dualize the statement o Lemma 2.4 and 2.5. Exercise 2.18 : Let (/) be a comma object. Is it true that p, θ exhibits rit q? Exercise 2.19 : Prove that in at there is an isomorphism q X p θ Y Z lan よ よ = lan よ (, 1) or each : a unctor between small cateories, and the Yoneda embeddins よ : P, よ : P. Exercise 2.20 : Prove axiom 3 in at; write explicitly the isomorphism θ o axiom 3. Exercise 2.21 []: Prove that there is a Yoneda structure on the 2-cateory o posets (objects: posets seen as cateories; 1-cells: monotone unctions; 2-cells: the partial order relation on the set Pos(X, Y ) o monotone unctions). Describe explicitly よ : P; Prove directly axioms 1 and 2; Prove axiom 3 pretendin to inore that P is blatanly a unctor, i.e. prove directly that θ and ι are deined and invertible (=identities). Does axiom 4 hold? Do this without usin the description o Pos Exercise 2.22 []: Prove that the isomorphism α that render P a pseudo-unctor satisy the commutativity P (h) P ()P h P P (h) P (P P h) = (P P )P h

11 FORML TEGORY THEORY 11 (note that this is both borin and diicult: you cna only use the universal property). Exercise 2.23 : Prove 2.15 and 2.16 in a similar -ormal- way o Exercise 2.24 []: Let E be a initely complete cateory, and K = at(e) the 2- cateory o cateories internal to E. Recall the deinition o an internal prounctor [or94, 8.2.1, 8.4.3]; prove that there is an equivalence Pro E (, ) = Pro E (1, op ) Prove that this correspondence is natural in, (which covariance type is it?). We deine an internal ull subcateory o E an object S o K with an internal prounctor s : 1 S inducin a ully aithul unctor K (X, S) Pro E (1, ) via precomposition. a 1-cell : in K admissible when the prounctor correspondin to (/) lies in the essential imae o the unctor K ( op, S) Pro E (1, op ). call this (unique) 1-cell op S. Prove that K has a Yoneda structure when (, 1) := : [ op, S] is the mate o, and P := [ op, S]. What happens when E is an elementary topos and S = Ω E? What happens when E is a Grothendieck topos and S = N is the natural number object?. Reerences [éner] Jean énabou, Distributors at work, Lecture notes written by Thomas Streicher. [or94] Francis orceux, Handbook o cateorical alebra. 1, Encyclopedia o Mathematics and its pplications, vol. 50, ambride University Press, ambride, 1994, asic cateory theory. [Gra74] John W. Gray, Formal cateory theory: adjointness or 2-cateories, Spriner-Verla, erlin, 1974, Lecture Notes in Mathematics, Vol [Lor15] Fosco Loreian, This is the (co)end, my only (co)riend, arxiv preprint arxiv: (2015). [Str74] Ross Street, Fibrations and yoneda s lemma in a 2-cateory, Proceedins Sydney ateory Theory Seminar 1972/1973 (Greory M. Kelly, ed.), Lecture Notes in Mathematics, vol. 420, Spriner, 1974, pp [Str80] Ross Street, Fibrations in bicateories, ahiers de topoloie et éométrie diérentielle catéoriques 21 (1980), no. 2, [SW78] R. Street and R. Walters, Yoneda structures on 2-cateories, J. lebra 50 (1978), no. 2, [Woo85] R.J. Wood, Proarrows II, ahiers de topoloie et éométrie diérentielle catéoriques 26 (1985), no. 2,