53

One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories: arXiv:1112.0040 (i.e. $(\infty,n)$-categories). This axiomatization includes several variants of $(\infty,n)$-category for finite n, such as what you mention in your question. In particular, when ...


33

As you point out, there are "inclusion" functors $\mathrm{Cat}_n\to \mathrm{Cat}_{n+1}$. These inclusion functors admit both left and right adjoints (in the sense of functors between $(\infty,1)$-categories). The right adjoint $r\colon \mathrm{Cat}_{n+1}\to \mathrm{Cat}_n$ is a kind of truncation functor which effectively removes all $(n+1)$-morphisms ...


33

There are several ways to interpret the homotopy hypothesis. Strictly speaking, Grothendieck's homotopy hypothesis is not a theorem yet: Grothendieck stated it in a very precise way in the very first pages of Pursuing stacks, by defining explicitly a notion of weak $\infty$-groupoid and by associating functorially to any space $X$ its weak higher groupoid $\...


29

For me this result fits in a context of other results that give complete algebraic invariants for homotopy types. The broad program sometimes goes under the rubric Whitehead's algebraic homotopy program. If we define a homotopy $n$-type (for $n \geq 1$) as an object of the localization of a suitable category of spaces (e.g., $Top$ or simplicial sets) with ...


25

Easy. Who's gonna write it? JDJ (Johan de Jong) has written almost the entire stacks project himself.


25

Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where $C$ is a small $\infty$-category, $R=\{f_i\colon X_i\to Y_i\}$ is a set of maps in $\mathrm{PSh}(C)=\mathrm{Fun}(C^\mathrm{op}, \mathrm{Gpd}_\infty)$, and $\...


20

The question used the phrase "still needed." This is a very loaded term, and the answer that you give will depend very strongly on how you interpret it. If, as Dylan does, we interpret this as asking whether some theorems make it mathematically necessary to use one of these strict models, I suspect that the answer is ultimately no. My reason is a little ...


18

Yes, this has been achieved in some sense. There is a (unpublished and possibly not yet written) work of Gaitsgory and Lurie where they propose an answer to this question. Given a split reductive group scheme $G$ over $Spec(\mathbb{Z})$ they use a version of the geometric Satake correspondence to construct a stable infinity category which is linear over the ...


18

Let me try to address the bulleted questions and simultaneously advertise the G-R book everyone has mentioned. Since the main question was about literature, I could also mention Drinfeld's article "DG quotients of DG categories," which nicely summarizes the state of the general theory before $\infty$-categories shook everything up. However, it doesn't ...


18

My colleague Dylan answered first (I keep telling him not to spend too much time on this toy :) but I both agree and disagree with his "Yes of course". The same words are used with different meanings and implications in the point-set and infinity category setting. So the word "convenient" has correspondingly different meanings! With the meaning of words in ...


17

Dominic Verity and I have been working to develop model independent foundations of $(\infty,1)$-category theory. Our aim at present isn't to cover all models of $(\infty,1)$-categories but only the "best-behaved ones" (which include quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets). The benefit to restricting to ...


17

In fact what you need is that your ∞-category is additive (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories are additive, so I'll just prove it in this case. Step 1: The pullback $X\times_Z Y$ is equivalent to the pullback $(X\times Y)\times_{Z\times Z}Z$. This fact ...


16

I don't actually see how to deduce the version of the classical SvKT on a set of base points $A$ directly from Lurie's version. It seems that to apply Lurie's theorem, we would need the stronger hypothesis that $A$ meets every path component of every finite intersection of open sets in $\mathcal{U}$ (which is reasonable, since the conclusion of Lurie's ...


16

A category object internal to simplicial sets is the same as a Segal space in which the Segal conditions hold on the nose instead of merely up to weak equivalence. In other words, a category is something whose nerve has unique horn fillers instead of merely contractible spaces of fillers. The above category objects generate a full sub-(relative category) ...


16

There are plenty of interesting dg-categories one can associate to a scheme. From the point of view of six functor yoga, these should be viewed as "categories of coefficients" for cohomology theories. For example, the derived category of quasi-coherent sheaves (or its various variants) is the category of coefficients for coherent cohomology, just as the ...


16

According to the about page of Kerodon: Kerodon is an online textbook on categorical homotopy theory and related mathematics. It currently consists of a single chapter, but should grow (slowly) over time. It is modeled on the Stacks project, and is maintained by Jacob Lurie. I think this may answer your question?


15

It appears that Geoffroy Horel has solved this problem completely: Geoffroy Horel, A model structure on internal categories in simplicial sets, Theory and Applications of Categories 30 No. 20 (2015) pp. 704–750 (journal page, arXiv:1403.6873)


15

At the risk of starting some kind of (un?)civil war, let me expand on my comments. First and foremost, let's address the interpretation of the question. The OP asks "do we need a model category of spectra". If we interpret need to mean "without a model category of spectra we are unable to prove our favorite theorems and make our favorite computations" I ...


15

The original question has been answered in the sense that there are people who are confident to prove every statement about spectra they care about without recourse to models or model categories of spectra. (That they might argue with statements only known due to specific manipulations of simplices in the model of quasi-categories is a different question as ...


14

As everybody's said, there's an obvious thing to do. As Yosemite Sam cites, it's done in Section 13 of the ArXiv version of DAG I -- you think of chain complexes as enriched over simplicial sets via Dold-Kan, and then apply the nerve construction. But there's an explicit thing you can do for any dg category, and I find it useful because it's given in terms ...


14

See the recent paper Lee Cohn, Differential graded categories are k-linear stable infinity categories, arXiv:1308.2587 where a proof has been written down. The precise statement is that the underlying $(\infty,1)$-category associated to the Morita model structure on dg-categories over $k$ (where fibrant objects are karoubian pretriangulated dg-categories) ...


14

The answer to Q1 is indeed yes. The construction you describe (category of operators followed by coherent nerve) gives a functor from the category of fibrant colored simplicial operads to the category of $\infty$-operads (in the sense of Lurie). This functor preserves weak equivalences and induces an equivalence on the level of homotopy categories. The ...


14

First, let me remark that your question does not seem to be about the homotopy hypothesis, but about rectification. More specifically, the homotopy hypothesis concerns the question of whether (some version of) weak globular groupoids is equivalent to, say, Kan complexes, while the rectification problem you mention asks whether weak globular groupoids are the ...


14

The answer is yes: see the paper of Chu-Haugseng-Heuts, "Two models for the homotopy theory of ∞-operads", arXiv:1606.03826. In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial operads", arXiv:1109.1004) proved a Quillen equivalence between simplicial operads and dendroidal sets. In the paper of Cisinski and Moerdijk that you link to, ...


13

For smooth projective varieties it is known that $$D^b_\infty Coh(X\times Y)\simeq Hom(D^b_\infty Coh(X), D^b_\infty Coh(Y))$$ compatibly with the composition you describe (note here $D^b_\infty Coh$ coincides with the $\infty$-category $Perf$ of perfect complexes). Hence your desired $(\infty,2)$-category is a full subcategory of the $(\infty,2)$-category ...


13

It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a recent result of Lennart Meier, a certain "double cosubdivision" of the Rezk classification diagram of a model category is a complete Segal space, so (by a result of Joyal and Tierney) we can take degreewise 0-...


13

Have you looked in the paper by Jean-Marc Cordier and myself: Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1 - 54? We defined a simplicial set of coherent natural transformations between two simplicial functors $F,G:C\to D$. That may be useful as an intermediate setting. There are 'rectification' results if $D$ is complete/...


12

Mike Shulman has a good discussion of what exactly needs to be shown in his Michael Shulman, "The univalence axiom for inverse diagrams" (arXiv:1203.3253) The issue is that in the type theory the classification of objects is by strict pullbacks. So roughly the problem is that one has to show the object classifier of a random $\infty$-topos is, when the ...


12

Here are a few observations... I think there exist stable infinity categories that are not the dg-nerve (resp. $A_\infty$-nerve) of a dg-category (resp. $A_\infty$ category). In particular, the category of spectra should not arise in this way. I think Keller has a paper on differential graded categories that answers this question; he notes at some point ...


12

More generally, the issue with such interpretation is that substitution in type theory is interpreted by pullback in category theory, and substitution in ordinary type theory preserves all type-theoretic operations strictly and functorially, so we need some model for an $(\infty,1)$-category in which pullback has these properties. This is the purpose of the ...


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