The infinity-categories tag has no wiki summary.

**17**

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**2**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**4**answers

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**

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**0**answers

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**

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**1**answer

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

**2**answers

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**

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**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**4**answers

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

**1**answer

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

**4**answers

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 ...