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17
votes
2answers
546 views

Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...
6
votes
1answer
579 views

$(\infty, 1)$-Yoneda embedding via the Grothendieck construction

Let $C$ be a quasi-category. Then there is an imbedding $$ C^{op} \times C \to \mathrm{Kan}$$ where $\mathrm{Kan}$ is the quasi-category of Kan complexes. This is essentially constructed in Lurie's ...
9
votes
3answers
853 views

how to make the category of chain complexes into an $\infty$-category

I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes. Has anyone ever written down ...
18
votes
0answers
634 views

$\infty$-topos and localic $\infty$-groupoids?

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales). For the record, this is proved by, starting ...
10
votes
2answers
604 views

Does the classification diagram localize a category with weak equivalences?

Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)_n$ is the nerve of ...
5
votes
1answer
368 views

Presheaves on a complete Segal space

Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which ...
11
votes
2answers
1k views

What is a symmetric monoidal $(\infty,n)$-category?

This question arose from reading Jacob Lurie's "Classification of topological field theories" paper. In that paper, he uses complete $n$-fold Segal spaces as a model for $(\infty,n)$-categories, but ...
4
votes
0answers
199 views

Infinity-groupoid on the etale site of a scheme.

Let $X$ be a topological space. We may naturally associate to $X$ a simplicial set which is a Kan complex. This in turn gives rise to the fundamental $\infty$-groupoid associated to $X$. From this ...
4
votes
0answers
371 views

New model Structure on $E_{\infty}$-algebras?

Let $\mathbf{sSet}$ be the category of simplicial sets. Is it possible to put a new model structure on $\mathrm{E}_{\infty}$-algebra (of simplicial sets) such that the weak equivalences and fibrations ...
9
votes
2answers
1k views

Semi-simplicial versus simplicial sets (and simplicial categories)

Hi, Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving ...
22
votes
1answer
11k views

If I want to study Jacob Lurie's books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...
5
votes
1answer
217 views

Slices of infinity sheaves

I know from classical category theory that if $C$ is a small category and $X$ is a presheaf, that there is a canonical equivalence $$Set^{C^{op}}/X \simeq Set^{\left(C/X\right)^{op}},$$ where $C/X$ is ...
6
votes
1answer
310 views

How equivalent are the theories of reduced and groupal $\infty$-groupoids?

I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this ...
15
votes
4answers
1k views

Proof that the homotopy category of a stable $\infty$-category is triangulated

I've been looking at various general strategies for proving that some category is triangulated, and Lurie manages to prove that a huge class of interesting examples of categories that we know about ...
5
votes
0answers
340 views

Cohesive ∞-Toposes

Say an $\infty$-topos $\mathbf{H}$ is cohesive if its global section geometric morphism $\Gamma : \mathbf{H} \to \infty \mathrm{Grpd}$ admits a further left adjoint $\Pi$ and a further right adjoint ...
10
votes
1answer
326 views

Is the model category of Complete Segal Spaces right proper?

Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories". Has ...
1
vote
2answers
650 views

What does coinfinite mean?

I'm reading up on maximal sets and the word "coinfinite" pops up in the first sentence. I tried searching on Wolfram Mathworld as well as Google, but nothing concrete has come up. What does it mean ...
18
votes
1answer
1k views

The Dold-Thom theorem for infinity categories?

Let $\mathcal{M}$ denote the category of finite sets and monomorphisms, and let $\mathcal T$ denote the category of based spaces. For a based space $X \in \mathcal T$, one has a canonical funtor $S_X ...
4
votes
1answer
493 views

Fibrations of Simplicial sets

Hello, Maybe it is too vague a question, but I would like to ask if anybody could say some explanatory words about the importance (for infinity category study) of studying all the kinds of fibrations ...
3
votes
2answers
389 views

(∞,1) vs Category weakly enriched over spaces

What is the difference between: ($\infty,1$) categories - in which have for two objects you have an ($\infty,0$) category of morphisms (i.e. a space of morphisms) and categories weakly enriched ...
5
votes
1answer
608 views

Spectra and localizations of the category of topological spaces

Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces using some kind of localization combined with other categorical ...
11
votes
1answer
619 views

Stable ∞-categories as spectral categories

Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an ...
6
votes
2answers
1k views

The Yoneda Lemma for (oo,1)-categories?

According to this page on the nLab, it is currently unclear whether or not the entire Yoneda lemma generalizes to $(\infty ,1)$-categories rather than just the Yoneda embedding. Have there been ...
9
votes
1answer
617 views

Cyclic spaces and S^1-equivariant homotopy theory

I'm trying to understand the relationship between cyclic spaces and S1-equivariant homotopy theory. More precisely, I only care about S1-spaces up to equivalence of fixed point spaces for the finite ...
8
votes
4answers
1k views

Categories which are not compactly generated

Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear ...
7
votes
1answer
311 views

Stable presentable categories as module categories

There is a theorem of Schwede and Shipley which classifies categories of modules over an A∞ ring spectrum as those stable presentable (∞,1)-categories with a compact generator. Suppose I ...
21
votes
4answers
2k views

(∞, 1)-categorical description of equivariant homotopy theory

I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...