MILY RIHL: -CTGORIS FROM SCRTCH (TH SIC TWO-CTGORY THORY) GRIL C. DRUMMOND-COL I m trying to be really gentle on the category theory, within reason. So in particular, I m expecting a lot o people won t have seen a two-category at all. You guys may know too much to be an ideal practice audience, but i you see anything. The goal o the talks this week will to be give a precise account o the development o dimensional categories. We ll talk about limits and colimits, adjunctions, ibrations, modules (prounctors), the Yoneda lemma, and Kan extensions. That will be the scope o the basic theory. There will be a number o instantiations o the categories I ll consider, but this will be model-independent, it will apply in a number o contexts but will be agnostic to which context. I ll call these -categories, this is a technical term that I ll explain, they live in an -cosmos. This won t be an axiomitization o the -categories, but o the things we need to prove theorems with them. This will be optimized or ( 1)-categories, which are roughly -dimensional categories such that morphisms above dimension 1 are invertible. There are many models o this. Some models include quasicategories (Joyal), complete Segal spaces (Rezk), Segal categories (Simpson), marked simplicial sets (Verity). We ll see that the development o the category theory is preserved and relected by nice unctors that change models. You won t need to know what any o these things mean. The way this works, -categories and the unctors between them live in something called the homotopy 2-category. This will be a strict two-category comprised o objects called -categories, which I ll denote,, C. morphism will be an ( -)unctor which I ll write. The two-morphisms or 2-cells will be natural transormations. One example to keep in mind is categories, unctors, and natural transormations. So what can we do in a 2-category? It has an underlying ordinary 1-category. We also have composition o 2-cells, which compose in two dierent directions. I I m given a 2-cell α g and β g h, then I get β α rom h. So you have identity two-cells so that the morphisms are the objects o a category themselves. This is vertical composition. I also have horizontal composition, i I have α g and γ h k with the targets o the irst equal to the sources o the second to get γα rom h to kg. This is pasting, I can compose vertically and then horizontally or horizontally and then vertically, then the compositions agree. More generally, i I have a diagram with two-cells, all consistently oriented, I get a unique composite. I claim that this is a good ramework to develop our basic category theory. Deinition 0.1. n adjunction between -categories consists o a pair o 1- categories,, a pair o unctors u and, and a pair o natural 1
2 GRIL C. DRUMMOND-COL transormations η id u and ϵ u id which satisy η ϵ = id and similarly or the other composition. This can be deined internally internally or any kind o -category. So let me show some o what you can do. Proposition 0.1. djunctions compose. I I have C and then we can deine a 2-cell or and we can paste together. I get the composite o the identities rom drawing a diagram like this: C C η ϵ η ϵ You can also deine an equivalence Deinition 0.2. n equivalence consists o a pair o -categories and, a pair o unctors u and, and a pair o natural isomorphisms id u and u id. xercise 0.1. Show that any equivalence can be promoted to an adjoint equivalence by changing one o the 2-cells. Corollary 0.1. ny equivalence is both a let and right adjoint. Proposition 0.2. Given an adjunction between and and equivalences and, then there is an adjunction between and. The proo is that we can make and adjoint in either direction and likewise or. xercise 0.2. I is let adjoint to u and then is let adjoint to u. I and are adjoint to u then, and inally and is an equivalence then so is.
MILY RIHL: -CTGORIS FROM SCRTCH (TH SIC TWO-CTGORY THORY)3 The homotopy 2-category in which we ve been working is the quotient o an -cosmos. Let me motivate this a bit. I you want to do the homotopy theory o something, you might hope that it lives in a simplicial model category. You can say that two homotopy theories are the same i their model categories are equivalent in some sense. I ll show that a good ramework to do our kind o theory is or something enriched over quasicategories, not Kan complexes. We actually won t need those details. Deinition 0.3. n -cosmos is a simplicially enriched category, it s something that has objects (an object is called an -category), homs between objects which are quasicategories, a particular sort o simplicial set, and a speciied class o maps called isoibrations. This should satisy axioms. Let me pause and say that there are canonical equivalences, that is an equivalence i and only i map(x, ) map(x, ) is an equivalence o quasicategories. Trivial completeness K has (simplicially enriched) terminal object 1, pullbacks o isoibrations, cotensors by simplicial sets, meaning that with U a simplicial set and an -category then U is my cotensored object. isoibrations are closed under composition, contain isomorphisms, and are stable under pullback. Further, i U i V is an inclusion o simplicial sets and p is an isoibration then V U U V is an isoibration. We call an isoibration that is a weak equivalence a trivial ibration. coibrations there exists a lit. There are some consequences trivial ibrations It ollows rom the deinitions that trivial ibrations are closed under composition, contain isomorphisms, are stable under pullbacks, and V U U V is a trivial ibration i either p is a trivial ibration or i is a trivial coibration in the Joyal model structure. actorization nother consequence, we can actor any X as a section o a trivial ibration ollowed by an isoibration. This is what goes into the proo o Ken rown s lemma. representable isoibrations I p is an isoibration then or all X, map(x, ) is a ibration onto map(x, ). We can have the optional axiom o Cartesian closure, that map(, C) map(, C ) map(, C ). Ret me remind you that a quasi-category is a simplicial set with illers or inner horns Λ n,k n There is a model category structure on simplicial sets or quasi-categories by Joyal such that the coibrations are monomorphisms, ibrant objects are quasi-categories,
4 GRIL C. DRUMMOND-COL ibrations between quasi-categories are isoibrations (so I have liting problem solutions as ollows) Λ n,k 0 where I is the pushout o n I 1 [02] [13] 1 3 quivalences are simplicial homotopy equivalences using I, it s given by a pair o maps and g so that there exists 7777777 id p 0 7 I g and likewise or. How about some examples. We have these or categories, quasicategories, or complete Segal spaces, Segal categories, naturally marked simplicial sets, each o which other than the irst is a model o (, 1)-categories. ach has a model category structure, all o which are equivalent. Taking isoibrations as ibrations, then weak equivalences are the equivalences in my cosmos. There are a ew other examples. Certain models o ( n)-categories are objects in an -cosmos; similarly higher complete Segal spaces. Finally, i is in an - cosmos then K/ is a slice cosmos whose objects are isoibrations with codomain. This axiomatizes what is needed or their basic homotopy theory. Now we can orget most o this. I want to now tell you about the quotient. Deinition 0.4. The homotopy 2-category K 2 o an -cosmos K has as its objects the -categories. The morphisms will be vertices in the mapping quasicategory, so that the 2-category and the -cosmos have the same underlying category. 2-cell between, g is represented by a 1-simplex in the quasicategory. We say that two parallel 1-simplices represent the same 2-cell i and only i they are connected by a 2-cell whose third ace is degenerate. There s a slicker way at this deinition. quivalently, there s a unctor rom quasicategories to categories which takes a quasicategory to its homotopy category. Then hom are ho(map(, )). That s the sense in which this is a quotient o the -cosmos. Let me now tell you the most important result o today. We have a priori two distinct notions o equivalence. Proposition 0.3. I I have a unctor in an -cosmos K, then the ollowing are equivalent: p 1
MILY RIHL: -CTGORIS FROM SCRTCH (TH SIC TWO-CTGORY THORY)5 (1) is an equivalence in the -cosmos K (2) is an equivalence in K 2. So we can work 2-categorically and capture the right thing. For 2 to imply 1, we need that i is an equivalence in K and in K 2 then is an equivalence. The proo uses that I is a path object or. Now i has an equivalnec inverse g, then g id, g id, so then by 2 out o three we know that by 2 out o 3, is an equivalence in K. I i apply the homotopy category unctor to the equivalence map(x, ) map(x, ), I get an equivalence hom(x, ) hom(x, ). Then by the Yoneda lemma, a representable equivalence is an equivalence in K 2. That s a good place to stop.