Schreiber Seminar on (∞,1)-Categories and ∞-Stacks

A topic schedule and references for a seminar on basic (∞,1)-category theory, held Winter 2009.

For more basic background see the previous Course on sheaves and stacks. For applications see the following Seminar on derived differential geometry.

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Motivation and aim

Here a word on the motivation of $(\infty,1)$-category theory for an audience familiar with model category theory, and the statement of a result that may serve as a guiding light for the development of the theory here.

We have seen the notions of

• (simplicial) model category

• Bousfield localization.

• model structure on simplicial presheaves.

A central result putting these concepts together is Dugger’s theorem:

Model category theoretic statement

Every combinatorial simplicial model category arises, up to Quillen equivalence, as a left Bousfield localization of the global model structure on simplicial presheaves over some category $C$.

One may ask for the intrinsic of this statement. A point of view that explains the technology of model categories, their Quillen equivalences etc. as the presentation of something conceptually more natural: Dugger’s theorem can be read in a precise sense as saying that combinatorial model categories provide a generators and relations presentation of (∞,1)-categories: a notion of category where hom-sets are refined to hom-Kan complexes.

Intrinsic statement

Every presentable (∞,1)-category arises as the (∞,1)-categorical localization of an (∞,1)-category of (∞,1)-presheaves.

It turns out that most of ordinary category theory has generalizations to (∞,1)-category theory and that all constructions in model category theory are models for this: concrete realizations of something that is more intrinsically defined.

Outline

Topics

Here is a bare list of possible topics. The items are repeated with background information and pointers to the literature below.

Basics of $(\infty,1)$-category theory

1. model structure on sSet-categories

2. quasi-category

   • Kan complex

   • fibrations of quasi-categories

3. relation between quasi-categories and simplicial categories

   • homotopy coherent nerve

   • (model structure for quasi-categories)

4. basic notions of (∞,1)-category theory

   • homotopy category of an (∞,1)-category

   • presentation by a category with weak equivalences: Dwyer-Kan localization

   • (∞,1)-functor

   • (∞,1)-category of (∞,1)-functors

   • (∞,1)-category of (∞,1)-categories

   • the archetypical $(\infty,1)$-category: ∞Grpd

   • opposite quasi-category

   • (∞,1)-category of (∞,1)-presheaves

5. (∞,1)-Grothendieck construction

   • Grothendieck construction

     • category fibered in groupoids

   • right fibration

   • universal fibration of (∞,1)-categories

Universal constructions in $(\infty,1)$-category theory

1. limit in a quasi-category

   • terminal object in a quasi-category

   • join of quasi-categories

   • homotopy limit

2. Cartesian fibration / (∞,1)-Grothendieck construction

3. adjoint (∞,1)-functor

4. localization of an (∞,1)-category/reflective (∞,1)-subcategory.

$(\infty,1)$-Sheaf and topos theory

1. (∞,1)-category of (∞,1)-sheaves

   • descent, Cech nerve, hypercover, hypercompletion, Whitehead theorem

2. presentable (∞,1)-categories and Dugger's theorem

References

A semi-technical survey of central aspects of $(\infty,1)$-category theory is in the section

• Perspective

of

• André Joyal, The theory of Quasi-Categories and its Applications notes from a lecture at Simplicial Methods in Higher Categories (pdf)

A survey with an eye towards the description of ∞-stack (∞,1)-toposes is the introduction of

• Jacob Lurie, Higher Topos Theory .

Topics and literature

Basic notions of $(\infty,1)$-category theory

Idea

There are several models for the notion of (∞,1)-category. Two of them are

• quasi-categories– these are simplicial sets satisfying a condition slightly weaker that that of a Kan complex: where the latter ensures that all cells may be composed and have inverses, the weaker condition requires on 1-cells only that they may be composed.

• Kan complex-enriched categories $C$ – we think of a $(k-1)$-simplex in the hom-object $C(X,Y)$ as a k-morphism between $X$ and $Y$.

  By the nature of ordinary enriched category theory the composition of such k-morphisms along objects is strictly unital and strictly associative, which makes this a semi-strict (∞,1)-category.

In practice one often passes back and forth between these two realizations, as convenient, using the homotopy coherent nerve $N : sSet Cat \to sSet$ and its left adjoint that establish the relation between quasi-categories and simplicial categories.

All the basic notions of category theory have pretty straightforwards analogs in both models, but some are more immediate in one model than in the other. For instance (∞,1)-functors are naturally formulated on quasi-categories, while hom-spaces are directly read off from $SSet$-enriched categories.

Topics

1. model structure on sSet-categories

2. quasi-category

   • Kan complex

   • fibrations of quasi-categories

3. relation between quasi-categories and simplicial categories

   • homotopy coherent nerve

   • (model structure for quasi-categories)

4. Basic notions of (∞,1)-category theory

   • homotopy category of an (∞,1)-category

   • presentation by a category with weak equivalences: Dwyer-Kan localization

   • (∞,1)-functor

   • (∞,1)-category of (∞,1)-functors

   • (∞,1)-category of (∞,1)-categories

   • the archetypical $(\infty,1)$-category: ∞Grpd

   • opposite quasi-category

   • (∞,1)-category of (∞,1)-presheaves

5. (∞,1)-Grothendieck construction

   • Grothendieck construction

     • category fibered in groupoids

   • right fibration

   • Cartesian fibration

   • universal fibration of (∞,1)-categories

References

The notion of quasi-category appeared first in

• Michael Boardman, Rainer Vogt, Homotopy invariant algebraic structures on topological

  spaces_, Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, 1973.

under the name weak Kan complex. Its role as a carrier of (∞,1)-category theory was understood by Andre Joyal, as exposed in

• André Joyal, The theory of Quasi-Categories and its Applications notes from a lecture at Simplicial Methods in Higher Categories (pdf).

…more goes here…

One procedure for turning a quasicategory into a simplicial category is described in:

• HTT, section 1.1.5

A different, equivalent, construction is in

• Dan Dugger, David Spivak, Rigidification of quasi-categories (arXiv:0910.0814)

• Dan Dugger, David Spivak, Mapping spaces in quasi-categories (arXiv:0911.0469)

Universal constructions in $(\infty,1)$-category theory

Idea

A crucial point of (∞,1)-category theory is that all the universal constructions known from category theory generalize to this context.

Since a limit in an ordinary category is a universal cone it is straightforward to say what a limit in a quasi-category is, once we have a notion of cone in that context. It turns out that this is neatly modeled by the notion of join of simplicial sets – a natural monoidal structure on SSet induced simply from the ordinal sum operation on the augmented simplex category.

Using the corresponding notion of join of quasi-categories we can speak of over quasi-categories and then define the limit over an (∞,1)-functor $F : D \to C$ as the (quasi-categorical) terminal object of the over quasi-category $C_{/F}$.

Such $(\infty,1)$-limits and colimits are what is modeled by homotopy limits and homotopy colimits in model category theory.

… Cartesian fibration… adjoint functor as cograph of a functor… Cartesian fibration over interval…localization by reflective $(\infty,1)$-subcategories…

Topics

1. limit in a quasi-category

   • join of quasi-categories

   • terminal object in a quasi-category

   • homotopy limit

2. Cartesian fibration / (∞,1)-Grothendieck construction

3. adjoint (∞,1)-functor

4. localization of an (∞,1)-category/reflective (∞,1)-subcategory.

References

Joins of quasi-categories are discussed

• Joyal, chapter 3

• HTT, section 1.2.8.

An equivalent variant is discussed in

• HTT, section 4.2.1.

Limits in quasi-categories are discussed

• Joyal, page 159

• HTT, section 4.

Adjoint $(\infty,1)$-functors are the topic of section 5.2 in that book.

$\infty$-Sheaf and topos theory

Idea

We have seen that it is straightforward to define the (∞,1)-category of (∞,1)-presheaves on an (∞,1)-category $C$:

$$PSh(C) := Func(C^{op}, \infty Grpd) \,.$$

We may then define an (∞,1)-category of (∞,1)-sheaves on $C$ to be a $(\infty,1)$-category reflectively embedded into $PSh(C)$, i.e. such that

$$Sh(C) \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} PSh(C)$$

is a full and faithful (∞,1)-functor with a left exact functor left adjoint (∞,1)-functor.

Topics

1. (∞,1)-category of (∞,1)-sheaves

   • descent, Cech nerve, hypercover, hypercompletion, Whitehead theorem

2. presentable (∞,1)-categories and Dugger's theorem

References

Categories of $(\infty,1)$-presheaves are the topic of section 5.1 of

• Jacob Lurie, Higher Topos Theory .

The theory of localizations of these is in sections 5.2.7 and 6.2.1 in

• Jacob Lurie, Higher Topos Theory .

General References

An accesible development of basics of quasi-category theory designed to serve as course notes is

• André Joyal, The theory of Quasi-Categories and its Applications notes from a lecture at Simplicial Methods in Higher Categories (pdf)

This is meanwhile developing in a textbook, which however is not yet available. But Joyal has begun working on expositional material here:

• André Joyal, The theory of quasi-categories

A pretty comprehensive development of the technology of (∞,1)-category theory and its sheaf and topos theory is in the book

• Jacob Lurie, Higher Topos Theory ,

which is however less expositional.

Quasi-categories originally appeared – under the term weak Kan complex – in

• Michael Boardman, Rainer Vogt, Homotopy invariant algebraic structures on topological

  spaces_, Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, 1973.

• Rainer Vogt, Homotopy limits and colimits, Math. Z., 134, (1973), 11?52.

but the insight that “there is category theory for quasi-categories” was pointed out only later by Joyal.

After a long while in which no generally good model for higher category theory seemed in reach, there is now a plethora of them available.

A survey of the relation between four different models for (∞,1)-categories is

• Julie Bergner, A survey of $(\infty,1)$-categories (arXiv)

Here we concentrate on two models: quasi-categories and simplicial categories. Their relation is the topic of HTT, section 1.1.5. The homotopy coherent nerve relating these in one direction goes back to

• J.-M. Cordier, Sur la notion de diagramme homotopiquement cohérent, Cahier Top. et Geom. Diff. XXIII 1, 1982, 93-112

A review of the construction is in

• Vivek Dhand, The simplicial nerve of a simplicial category (pdf)

This and its left adjoint is also discussed in section 1.1.5 of

• Jacob Lurie, Higher Topos Theory

An alternative, equivalent but possibly more insightful, construction of the left adjoint is the topic of

• Dan Dugger, David Spivak, Rigidification of quasi-categories (arXiv:0910.0814)

• Dan Dugger, David Spivak, Mapping spaces in quasi-categories (arXiv:0911.0469)

When it comes to (∞,1)-category theory proper, its universal constructions such as limits, adjunctions, Grothendieck construction, etc. pretty much the only one-and-a-half source available are

• André Joyal, unpublished book

and sections 4 (limits) and 5.2 (adjunctions) of

• Jacob Lurie, Higher Topos Theory .

For the theory of localizations of $(\infty,1)$-presheaf categories there is section 5.2.7 and section 6.2.1 in that book.