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Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" of an (infinity,1)-category. Also, "model" in quotes means the English word model, whereas without quotes it has do do with model categories.

At the very beginning of Lurie's higher topos theory, he mentions that there is a theory of $(\infty,1)$-categories that can be directly constructed by using model categories.

What I'd like to know is:

Where can I find related papers (Lurie mentions two books that are not available for download)?

How dependent on quasicategories is the theory developed in HTT? Can the important results be proven for these $(\infty,1)$-model-categories by proving some sort of equivalence (not equivalence of categories, but some weaker kind of equivalence) to the theory of quasicategories?

When would we want to use quasicategories rather than these more abstract model categories?

And also, conversely, when would we want to look at model categories rather than quasicategories?

Does one subsume the other? Are there disadvantages to the model category construction just because it requires you to have all of the machinery of model categories? Are quasicategories better in every way?

The only "models" of infinity categories that I'm familiar with are the ones presented in HTT.

David White
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Harry Gindi
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5 Answers5

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You can't produce every ($\infty$,1)-category from a model category. The slogan is that every presentable ($\infty$,1)-category comes from a model category, and every adjoint pair between such comes from a Quillen pair of functors between model categories. The paper by Dugger on Universal model categories works out this formalism from the point of view of model categories. (A companion paper shows that a large class of model categories (the combinatorial ones) are "presentable" in this sense.)

(I say it's a slogan, but I'm sure it's a theorem; I just don't have a reference for you.)

Presentable ($\infty$,1)-categories are special among all ($\infty$,1)-categories; in particular, they are complete and cocomplete.

For instance, you can define the notion of ($\infty$,1)-topos in terms of model categories, since ($\infty$,1)-topoi are presentable, and morphisms between such are certain kinds of adjoint functor pairs.

Charles Rezk
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  • Is there any reason to generalize past presentable $(\infty,1)$ categories other than for the sake of generalization? I presume so, but I don't have the background to know why we'd want to. – Harry Gindi Dec 12 '09 at 15:46
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    Well, if you're interested in small ($\infty$,1)-categories, you need something more general than model categories. Small ($\infty$,1)-categories are (almost) never presentable. – Charles Rezk Dec 12 '09 at 15:50
  • We can't use the usual universes trick is what you're getting at? Or do you mean small in the sense of not being complete and cocomplete? Why would we want to look at small $(\infty ,1)$ categories (this question may be dumb)? If they're not small, we can't define a suitable $(\infty,2)$ category of $(infty,1)$ categories? – Harry Gindi Dec 12 '09 at 15:58
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    I dunno. Take a Lie group G. This is a topological group, so it is an ($\infty$,1)-category (use the "model" of ($\infty$,1)-categories as topologically enriched categories). It is small, and is not presentable. Lie groups are interesting, and one might want to think about "representations", which in some general formulation are functors from G to some fixed ($\infty$,1)-category C.

    You can think about this using model categories, if C is a topologically enriched model category (there's a model category of enriched functors $G\to C$). But it's a bit awkward as a general formalism.

     – Charles Rezk Dec 12 '09 at 16:20
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    My favorite small $(\infty,1)$-categories are those of perfect complexes on a scheme say, or more generally compact objects in a stable presentable $\oo$-category. Anytime you want to put finiteness conditions on your modules/representations/sheaves/spaces etc you leave the presentable world (though you can usually study them using presentable tools by say passing to Ind-categories). – David Ben-Zvi Dec 12 '09 at 20:22
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    Ordinary category theory would be pretty limited if it could only talk about locally presentable categories! How do you even say something like "C is complete" i.e. "C has limits indexed by small categories" if you don't know what a "small category" is? The same applies in the $(\infty,1)$-world. – Mike Shulman Dec 12 '09 at 20:44
  • Another reason to generalize past the locally presentable case is that you'd like an $(\infty,1)$-category of functors between two given $(\infty,1)$-categories. – Mike Shulman Dec 12 '09 at 20:45
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    Aren't categories of continuous functors between presentable \infty,1-categories presentable? (is "continuous" the problem? or is "locally" the point you were making?) – David Ben-Zvi Dec 12 '09 at 20:51
  • (This is getting a little off-topic, but are there people who use "presentable" and "locally presentable" non-synonymously?) – Reid Barton Dec 12 '09 at 21:44
  • @Reid: I presume so. Category theorists came up with the name "locally presentable" a while ago - they must have had a reason for distinguishing this from plain "presentable", no? – Charles Rezk Dec 12 '09 at 22:02
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    The problem with "presentable" is that there is a general notion of a "presentable object" in a category, namely one such that hom(x,-) preserves k-filtered colimits for some k. So "presentable category" obviously means "presentable object in Cat." The term "locally presentable category" indicates that it is not the category itself, but its objects, which are presentable. – Mike Shulman Dec 13 '09 at 03:31
  • @David: I think probably "continuous" is the problem, although I'm not sure what exactly you mean by it. – Mike Shulman Dec 13 '09 at 03:38
  • By continuous I mean colimit preserving (which seems to be the natural class of functors between presentable categories), and Lurie describes a presentable oo-category of continuous functors between presentable oo-categories -- was your comment an assertion regarding arbitrary rather than colimit preserving functors? – David Ben-Zvi Dec 13 '09 at 04:32
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    That's an unfortunate use of "continuous," because "continuous" is well-established in ordinary category theory to mean "(small) limit-preserving," while "cocontinuous" means "(small) colimit-preserving." Also, I would have thought the natural class of functors between locally presentable (∞-)categories would be the accessible functors, i.e. those preserving k-filtered colimits for some k. I can think of lots of natural functors on locally presentable (∞-)categories that are accessible but don't preserve arbitrary colimits. But anyway, I was thinking about arbitrary functors. – Mike Shulman Dec 14 '09 at 04:23
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    Just a remark: the slogan is Proposition A.3.7.6. of Higher Topos Theory. –  Jun 29 '15 at 01:25
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Here is an extended comment regarding Charles answer, including some more references.

In this paper, Theorem 2.5.9, it is shown that every model category (not necessarily a combinatorial one) has all limits and colimits. However, it is not hard to find examples of model categories whose underlying $\infty$-categories are neither presentable nor co-presentable. For instance, Isaksen's strict model structure on pro-simplicial sets. It is shown in the paper mentioned above that the underlying $\infty$-category of this model category is the pro category of spaces considered in Lurie's "Higher Topos Theory" Definition 7.1.6.1. The pro category of a large cocomplete and finitely complete $\infty$-category is complete and cocomplete but neither presentable nor copresentable.

In this paper, Proposition 1.5.1, it is shown that any Quillen pair between model categories (not necessarily combinatorial ones) gives rise to an adjoint pair of $\infty$-categories.

It seems plausible to me that the (underlying $\infty$-category of the) relative category of model categories and left Quillen functors between them, with weak equivalences taken to be the Quillen equivalences is equivalent to the $\infty$-category of complete and cocomplete $\infty$-categories and left adjoints between them. Restricting to combinatorial model categories would correspond to restricting to presentable $\infty$-categories in the image. The latter statement is probably already proven, whereas I am pretty sure the former is not.

David White
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Ilan Barnea
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    This is somewhat relevant: http://ncatlab.org/nlab/show/derivator#as_a_presentation_of_categories – David Roberts Aug 18 '15 at 00:32
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    An arXiv preprint from today, "Combinatorial model categories are equivalent to presentable quasicategories" by Dmitri Pavlov, is relevant here: https://arxiv.org/pdf/2110.04679.pdf – Bruno Stonek Oct 12 '21 at 09:37
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You might want to take a look at the responses to How to think about model categories?

Even if you only cared about presentable (∞,1)-categories, you still might prefer the general (∞,1)-categorical machinery to that of model categories, if only to speak of the (∞,1)-category of all presentable (∞,1)-categories. I don't know of any corresponding "model 2-category" of model categories (though one can see traces of what it ought to be in the work of Dugger).

However, model categories do have certain advantages. There are some model categories arising from algebra for which I don't know any alternative description of the associated (∞,1)-category, such as the model category of chain complexes of comodules over a Hopf algebra. Even when this is not the case, model categories are often more convenient for computation. For instance, I can write down a (∞,1)-categorical description of some group cohomology, but to compute it I'll probably write down a resolution of something, which is the prescription coming from model categories.

Community
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Reid Barton
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  • Would you mind elaborating on your Hopf algebra example? if the Hopf algebra is for concreteness functions on an algebraic group, aren't you describing sheaves on the algebraic stack BG, which have a natural oo-categorical description (eg as a limit, or more directly as comodules for a comonad)? – David Ben-Zvi Dec 12 '09 at 20:10
  • Oh, I was worried about the case where the Hopf algebra is not flat over the base ring, but I guess already in ordinary algebra that is considered a Bad Thing to Do. – Reid Barton Dec 12 '09 at 21:51
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There are four different models for $(\infty,1)$-categories: simplicial categories, Segal categories, complete Segal spaces and quasicategories. A survey of these and their equivalences is provided by Julia Bergner's A survey of $(\infty, 1)$-categories.

Alicia Garcia-Raboso
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    There are more than four. In addition to variants of simplicially enriched categories, Segal categories, and complete Segal spaces which use, say, topological spaces instead of simplicial sets, there are also $A_\infty$-categories. – Mike Shulman Dec 12 '09 at 20:46
  • Doesn't the fact that topological spaces and simplicial sets are Quillen equivalent imply that the variants you mention are essentially the same thing? – Alicia Garcia-Raboso Dec 12 '09 at 21:49
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    Well, yes, but all the models for $(\infty,1)$-categories are Quillen equivalent, so they are all "essentially the same thing." (-: Simplicially enriched and topologically enriched categories are certainly more closely related than either one is to quasicategories, but they are not identical either. For instance, every topologically enriched category is fibrant, which is not the case for simplicially enriched ones. – Mike Shulman Dec 13 '09 at 03:34
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There is a followup article, first in the 5-article series Derived Algebraic Geometry:

In particular,

The theory of stable ∞-categories is not really new: most of the results presented here are well-known to experts. There exists a sizable literature on the subject in the setting of stable model categories (see, for example, [27]). The theory of stable model categories is essentially equivalent to the notion of a presentable stable ∞-category, which we discuss in §15.

Zhen Lin
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Ilya Nikokoshev
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    I don't know much, but I think essentially similar to "Applications" chapter in http://hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf . I'll write more if I have time on the weekend :) – Ilya Nikokoshev Dec 12 '09 at 12:13