∞-categories Summer Mini-course at UT

Posted

Jul 22, 2020 09:53 PM

👉 Lecture 1 Jul 20, 2020

Goal of this mini-course: Enable participants to tackle introductory literature such as HTT by Lurie or HTHA by Cisinski.

Pre-requisites: Model categories and simplicial sets.

Format: Lectures with interspersed exercises.

1. Introduction

An infinity category is the notion of category interpreted in homotopy theory. We replace the usual categorical identities with homotopy [equivalences] and replace sets by homotopy types. Quasi-categories, defined below, offer a way of making this precise.

Recall the nerve functor given by

is fully faithful. Because of this, is equivalent to a full sub-category of , the essential image of , consisting of all simplicial sets isomorphic to the nerve of some category. There are various intrinsic descriptions of this essential image, one of which is via Segal conditions: simplicial sets such that

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Note: The essential image of a functor is the smallest replete subcategory of containing the image of . A replete subcategory is such that for any and any in , are also in .

For instance, for m + n = 1 + 1 this amounts to saying that the 2-simplices in the nerve of a category are pairs of composable morphisms

But while this condition characterizes all simplicial sets isomorphic to the nerve of a category, there are additional "category-like" simplicial sets not in the essential image. Consider for instance the category of CW-complexes. We can take it's nerve to obtain a simplicial set as above, or we could instead extract a simplicial set which encodes the homotopy theory of :

sets of composable morphisms with homotopies on every triangle

In general, there exists a functor from simplicially enriched categories to simplicial sets called the homotopy coherent nerve.

Note: more on the homotopy coherent nerve

The homotopy coherent nerve (or simplicial nerve) of a category enriched in simplicial sets includes information about the higher simplices included in the hom sets (whereas the ordinary nerve would only consider sequences of 1-morphisms). Just as the nerve of a category is defined by taking functors from categories , the simplicial nerm of a -enriched category is defined by taking for a special cosimplicial simplicially-enriched category which we now describe: for

the objects of are

the homs object from to is the simplicial set given by the nerve of the poset of subsets of that contain both

composition is given by union of subsets

The homotopy coherentt nerve is then defined as

Recall that the standard -simplex is the simplicial set defined as the functor The k-th horn of dimension n for is the simplicial subset of spanned by all faces except the -th. is an inner horn if . So for instance, for we have

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Definition. A quasi-category is a simplicial set with all inner horn fillers, i.e. such that for all with , there is a map extending .

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Proposition. A quasi-category is (the nerve of) a category if and only if the inner horn fillers are unique.

Certainly categories themselves are an example (two functions have a unique composition, e.g.).

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Definition. A functor of quasi categories is a map of simplicial sets, and a natural transformation is a functor .

In fact, the cartesian closed category of simplicial sets restricts to a cartesian closed category of quasi-categories. In this category, the morphism set from to is denoted as usual, but the quasi-category of morphisms is denoted

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Definition. A morphism in a quasi-category is called invertible if there exist such that and .

Exercise. Prove that there exists a simplicial set classifying invertible morphisms equipped with a left and right inverse, and that . Hint: is not a quasi-category. (Toggle to show solution)

The existence of means around any invertible there is a driagram as follows. Furthermore the existence of such a diagram is necessary and sufficient for t o be invertible.

So is the simplicial set

Quasi-category theory is a conservative extension of category theory, in that any quasi-categorical statement that can be interpreted in classical category theory reduces to a classical statement. For instance, quasi-categorical (co)limits in ordinary categories are ordinary (co)limits. Thus, -category theory also sheds light on ordinary category theory. For instance, it highlights the importance of the join construction, initial/final functors, smooth/proper functors, etc.

2. Coherence

The key difficulty in carrying out the quasi-category program lies in exhibiting coherence. What is coherence? The answer is more or less philosophical. It can manifest as universal properties or as assembling local constructions into global ones. Some examples in ordinary category theory are the following facts:

fully faithful essentially surjective functors are equivalences of categories

a natural transformation is invertible iff its constituent morphisms are so

On the other hand, for a functor of quasi-categories we must also specify compatibility of composition in with composition in .

An easier proposition is the following:

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Claim. Consider a functor of quasi-categories and for every 0-simplex (object) , invertible functions , then there exists a functor sending to , together with a natural transformation with constituent morphisms given by .

We will show this later using the structure of model categories, which are equipped with classes of fibrations, cofibrations, and weak equivalences, defined categorically via several lifting properties. Here is a preview of how the proof will go. Let be the simplicial set defined in the exercise above for classifying invertible morphisms equipped with a left and right inverse.

Proof. is a trivial cofibration in the quasicategory model structure. Let be the constant simplicial set corresponding to the -simplices of , then we have is a cofibration in the quasicategory model structure. Tomorrow, we will show this implies is a fibration, so we have the diagram on the right. ◼️

The quasicategory model structure is called the Joyal model structure. This proof illustrates a general principle: rather than assembling data piece by piece, we construct some space or quasicategory of globally assembled data.

👉 Lecture 2 Jul 21, 2020

3. Foundations

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Note: For any small category , its category of presheaves is the functor category .

In any presheaf category we have a cofibrantly generated Cisinski model structure, where we use an exact cyllinder to define equivalence relations on hom sets, and -anodyne extensions give us a left class, closed with respect to operations involving

For a class of morphisms let be the set of all morphisms with the right lifting property with respect to all maps in , and similarly let be all morphisms with the left lifting property. is said to be a left class if .

In a category with a notion of fibrations and a given class of such, a morphism is anodyne if it has the left lifting property against all of these.

Note: more on model structures

A model structure on a category is a choice of three classes of morphisms each closed under composition—weak equivalences , fibrations , and cofibrations —satisfying that

contains all isomorphisms and given a pair of composable morphisms such that two of are in , so is the third

and are weak factorization systems on , meaning that for each of these pairs , any morphism can be factored as for and such that is precisely the class of morphisms having the left lifting property respect to any morphism in and vice versa.

A fibrant object is an object that has a fibration to the final object in .

A trivial cofibration is an element of , and the definition of trivial fibration is similar.

If a morphism has the left lifting property respect to , we write .

Note: more on Cisinski's theory

🚨TO-DO

The Cisinski model strucutre is as follows

We let cofibrations be monomorphisms (the trivial fibrations are the same for all Cisinski model structures on )

The fibrant objects are such that lifts against -anodyne extensions.

Fibrations between fibrant objects are those which lift -anodyne extensions.

is a weak equivalence fibrant, is a bijection.

-equivalence relation coincides with homotopy relation on fibrant objects.

Observations: There are no explicit generators for trivial cofibrations, and -anodyne extensions trivial cofibrations.

Example

(equivalence relation on hom sets is the usual homotopy relation)

-anodyne extensions are generated by

This is the Kan-Quillen model structure.

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Theorem. For and , the generators for the -anodyne maps are the inner horn inclusions, and this gives the Joyal model structure.

The weak equivalences are called weak categorical equivalences and the -anodyne extensions are called categorical anodyne extensions. The generating cofibrations for the Joyal model structure are .

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Theorem. The fibrant objects in the Joyal model structure are presicely the quasi-categories.

For now we consider four weak factorization systems:

Monos

Trivial cofibrations

Categorical anodyne extensions

Inner anodyne extensions =

Trivial fibrations

fibrations

"naive fibrations''

inner fibrations

3.1 Two-variable Quillen adjunctions

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Definition. A two variable adjunction between categories is a triple of functors , such that naturally in .

How do two variable adjuntions interact with w.f.s?

Warm-up: For any ordinary adjunction and morphisms , ,

Similarly, if have pullbacks and has pushouts, then for any , ,

Exercise: Prove this! (Toggle to show solution)

Answer: Since this is an ordinary adjunction we have for each . In particular, the map on the left diagram corresponds uniquely to on the right and corresponds uniquely to . For the direction assume we have a lift on the left diagram, then by adjunction it corresponds uniquely to a . This map makes the right diagram commute by the adjunction hom isomorphism: on the right corresponds to on the left which corresponds to on the right. Same for the top triangle. The converse is identical.

Now for the second set of implications. Lets do first lift second lift. Lets draw the pushout square for and the pullback square for :

First notice that is simply the map induced by and similarly is simply the map induced by . Similarly is the map induced by (recall that is contravariant on ) and is the map induced by ( is covariant on ). The map on the second diagram is defined by taking the adjoints and of and on the first diagram. Now assume we have a lift in the first diagram. This induces an adjoint which we must show makes the right diagram commute. To show the maps and are equal it suffices to show their composites to and are, by the universal property of the pullback.

is adjoint to on the first diagram, which is equal to by commutativity of the first diagram, and this is adjoint to so the composites to agree.

similarly, is adjoint to which by commutativity of the first diagram equals , which is adjoint to , so the compositions to agree.

Thus a lift in the first diagram induces a lift in the second diagram, which shows the first . The other implications follow similarly by spamming adjointness and assuming commutativity of one of the diagrams, and I'll refrain from writing them for now because this paragraph has already gotten super long! ◼️

Example: (of ) The categorical anodyne extensions are given by

Exercise: Show that the fibrant objects in the Joyal model structure are quasi-categories.

Solution: The condition that satisfy right lifting against every inner horn inclusion is equivalent to the inner horn filler description defining a quasi-category.

What remains to do is to go in the other direction and show all quasi-categories are fibrant objects.

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Definition. For model categories , a two variable adjunction is a two variable Quillen adjunction if (1) preserves cofibrations in each variable separately (2) For any two cofibrations , , the map is a cofibration and moreover a weak equivalence if or is.

Proposition. For model categories , a two variable adjunction

is a two variable Quillen adjunction

is a two variable quillen adjunction.

Examples

A simplicial model category is precisely a two variable quillen adjunction

where is equipped with the Kan-Quillen model structure. More generally, a model category over a model category is a two variable Quillen adjunction

A monoidal model category is a model category with a two variable Quillen adjunction

E.g. with the Kan-Quillen or Joyal model structure, or sending . ("This is the only example that I have where all 3 categories are different.")

4. Applying Two Variable Lifting

We will often only have partial two variable Quillen adjuntions, i.e. missing or . In all cases, Quillenness is proved using generating sets.

"Optional" Exercise: Read and understand proof

Gabriel & Zisman 1967: Theorem 2.1 ("Very technical but once you figure out what they're doing it's very intuitive, at least in one direction.")

HCHA: Prop 3.2.3 ("Even more technical. Just if you wanna see very specifically how combinatorics comes into proving things about model categoreis")

Two very nice instances of producing different generating sets for a left class.

4.1. Joyal's theorem

One of the first papers to talk about quasi-categories, and talks about quasi-categories as a generalization of categories. Recommend reading this paper: very short, explicit, and self-contained.

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Theorem. (Joyal 2002) quasi-categories, and inner fibration (recall these are the maps which lift along the inner horns). For any diagram as the one here, such that is given by restriction of to is invertible there exists a lift.

Corollary. A quasi-category is a Kan complex iff all its morphisms are invertible.

4.2 The functors and

All due to Cisinski (3.5, 3.6, 3.9 in HCHA).

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Definition. Let be a category, then is the core of , i.e. the subcategory of invertible morphisms in .

With this at hand, we can define a core for quasi-categories.

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Definition. For a quasi-category, we define the core of by the pullback of . Equivalently, this is the largest subobject such that is a Kan complex.

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Definition. For a simplicial set and a quasi-category, we define by the pull-back of .

Where recall from last time that an underlined is the internal Hom object in the category of simplicial sets. We obtain a canonical inclusion from the square