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3
votes
1answer
282 views

Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology

Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this ...
8
votes
1answer
279 views

Loops and suspensions of higher categories

Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ ...
6
votes
1answer
235 views

Compact objects in undercategories and filtered colimits

Let $\mathcal{C}$ be a compactly generated presentable $(\infty, 1)$-category. Consider the functor $$ \Phi: \mathcal{C} \to \mathrm{Cat}_\infty, \quad x \mapsto (\mathcal{C}_{x/})^\omega,$$ that ...
3
votes
0answers
298 views

Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry. Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...
10
votes
2answers
568 views

What is a Homotopy between $L_\infty$-algebra morphisms

A $L_\infty$-algebra can be defined in many different ways. One common way, that gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
6
votes
0answers
162 views

Lex $\infty$-colimits

In the paper lex colimits, Garner and Lack gave a general characterization of "exactness properties" for categories (and enriched categories). A "lex-weight" is a weight for colimits whose domain ...
6
votes
2answers
296 views

Coequalizers in stable (infinity,1)-categories

I have read it claimed in several places that in a stable $(\infty,1)$-category, the coequalizer of parallel maps $f,g:X\to Y$ can be identified with the cokernel of $f-g$ (i.e. the pushout of the map ...
13
votes
2answers
673 views

$\infty$-categorical interpretation of type theory

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity ...
3
votes
1answer
210 views

Does Ind-completion commute with finite limits?

The broad and vague question is in the title. The more precise question is: Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with ...
10
votes
2answers
466 views

Correspondence between operads and $\infty$-operads with one object

Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose ...
38
votes
4answers
2k views

Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
12
votes
0answers
209 views

Constructible derived category and fundamental category

Introduction (may be skipped) Given a nice topological space $X$, the category of local systems (say over a field $k$) on it is equivalent to the category of representations of its fundamental ...
3
votes
1answer
180 views

Directed colimits of maps in a combinatorial model category

I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated ...
9
votes
0answers
419 views

The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...
3
votes
0answers
241 views

Simplicial localisation and infinity categories

If $(\mathcal{C},W)$ is a category with weak equivalences then we may naturally form its Dwyer-Kan simplicial localisation $L(\mathcal{C}, W)$. This is a simplicial category which naturally gives a ...
10
votes
1answer
352 views

Are $\infty$-topoi determined by their localic points ?

Hello ! If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an ...
16
votes
2answers
822 views

Internal categories in simplicial sets

Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories? Note that this ...
9
votes
2answers
1k views

Stable infinity categories vs dg-categories

What is the relation between dg-categories and stable $\infty$-categories? Given a dg-category one can form its dg-nerve and get a $\infty$-category (which will be stable if the dg-category is?). ...
3
votes
0answers
131 views

Right Notion of Localizing Subcategory in Quasicategory

Given a stable quasicategory $C$, what are the conditions required of a subcategory for it to descend to a localizing subcategory in $Ho(C)$? Is it enough for it to be reflective? Perhaps exact and ...
24
votes
2answers
1k views

generalisations of the Seifert-van Kampen Theorem?

I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available. My attention was ...
10
votes
0answers
271 views

Adjoint functor theorem for infinity categories

In HTT, a version of the adjoint functor theorem for (locally) presentable infinity categories is proven (Corollary 5.5.2.9). Is there a more refined version of this somewhere, which more closely ...
2
votes
0answers
207 views

On small generators of an infinity category

Suppose that $\mathcal{C}$ is an $\infty$-category with pullbacks and small coproducts. Suppose that $\mathcal{D}$ is a small subcategory for every object $C,$ the canonical map ...
18
votes
2answers
570 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
773 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 ...
10
votes
3answers
974 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
655 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 ...
12
votes
2answers
739 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
384 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 ...
12
votes
2answers
2k 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
208 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 ...
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 ...
29
votes
1answer
13k 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
219 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
318 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 ...
6
votes
0answers
352 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
346 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
699 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
502 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
406 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
617 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 ...
12
votes
1answer
658 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 ...
7
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
643 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
321 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 ...
22
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 ...