# Questions tagged [higher-category-theory]

For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

**39**

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### Non-examples of model structures, that fail for subtle/surprising reasons?

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...

**31**

votes

**3**answers

2k views

### How do you define (infinity,1) categories in Homotopy Type Theory?

One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...

**25**

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**2**answers

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### 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 ...

**12**

votes

**2**answers

2k views

### Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...

**11**

votes

**3**answers

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### Infinity-categories vs Kan complexes

It is known (cf. Lurie's book Higher Topos Theory for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan complexes, aka ...

**14**

votes

**0**answers

620 views

### Cohesive ∞-toposes for analytic geometry

There is a class of big ∞-toposes that come with a good supply of intrinsic notions of differential geometry and differential cohomology: called cohesive ∞-toposes (after Lawvere's cohesive toposes).
...

**9**

votes

**2**answers

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### What is descent data (of higher categories), conceptually?

First consider a scheme $X$ with an open cover $\mathcal{U}=\{U_i\}$. An object with descent data on $\mathcal{U}$ is a collection $(\mathcal{E}_i,\phi_{ij})$ where $\mathcal{E}_i$ is a quasi-...

**6**

votes

**1**answer

653 views

### What is a higher derived constructible sheaf

Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C_*(\Omega X,k)$ of ...

**13**

votes

**2**answers

555 views

### Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...

**6**

votes

**1**answer

216 views

### What's the (monoidal) image of a monoidal functor?

For an ordinary functor $F\colon \mathcal{C} \to \mathcal{D}$ of categories, there is a construction $\operatorname{im} F$, the image of $F$, which is again a category, and $F$ factors through that ...

**55**

votes

**4**answers

11k views

### Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...

**63**

votes

**6**answers

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### What are Jacob Lurie's key insights?

This question is inspired by this Tim Gowers blogpost.
I have some familiarity with the work of Jacob Lurie, which contains ideas which are simply astounding; but what I don't understand is which key ...

**82**

votes

**3**answers

7k views

### What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...

**26**

votes

**2**answers

4k views

### Derived Algebraic Geometry and Chow Rings/Chow Motives

I recently heard a talk about Chow motives and also read Milne's exposition on motives. If I understand it correctly, the naive definition of the Chow ring would be that it simply consists of all ...

**23**

votes

**3**answers

4k views

### Conjectures in Grothendieck's “Pursuing stacks”

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this ...

**19**

votes

**3**answers

2k views

### What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-...

**12**

votes

**5**answers

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### Computations in $\infty$-categories

Direct to the point.
Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that ...

**11**

votes

**4**answers

1k views

### Limits in an $(\infty,1)$-category

In ordinary category theory, the notion of limit in a category $C$ is usually formulated with a category (of indices) $J$ and a functor $F:J\to C$ (a diagram in $C$), and a limit of this diagram is ...

**15**

votes

**1**answer

290 views

### Descent of Higher categorical structures along geometric morphisms

Let $f: \mathcal{E} \rightarrow \mathcal{T}$ be a geometric morphism between two (Grothendieck) toposes (or maybe more generally a bounded geometric morphism between elementary toposes).
It is well ...

**13**

votes

**4**answers

1k views

### What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence ...

**10**

votes

**2**answers

2k views

### Are grothendieck universes enough for the foundations of category theory?

Grothendieck universes are equivalent to ZFC+a strongly inaccessible cardinal. This is low on the large cardinal axiom list. Is it enough to place category theory on a firm foundational basis, and how ...

**22**

votes

**3**answers

2k views

### Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...

**13**

votes

**2**answers

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 ...

**19**

votes

**0**answers

469 views

### A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...

**19**

votes

**0**answers

449 views

### Is the category Idem filtered?

I have recently been reading Lurie's "Higher Topos Theory", and come upon what I believe to be an erronous claim. However, the author goes to some pain as a result of that claim, and the error seems ...

**15**

votes

**1**answer

1k views

### Are there non-categorical notions in topos theory?

Suppose that $\mathcal{T}$ is an abstract $2$-category we know is equivalent to the $2$-category of Grothendieck topoi via some equivalence $$\phi:\mathcal{T} \to \mathfrak{Top},$$ and let $E$ be an ...

**12**

votes

**2**answers

2k 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 ...

**17**

votes

**0**answers

693 views

### Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title.
In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are $2$-functors....

**15**

votes

**3**answers

523 views

### Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categories?

Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences.
Let $\mathbf Q$ be the corresponding $\infty$-...

**13**

votes

**0**answers

222 views

### Localizing $\mathrm{CombModCat}$ at the Quillen equivalences

Let $\mathrm{CombModCat}$ be the category of combinatorial model categories with left Quillen functors between them. By Dugger's theorem and the appendix of Lurie's "Higher Topos Theory" it ought to ...

**13**

votes

**1**answer

666 views

### Difficulties with descent data as homotopy limit of image of Čech nerve

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ...

**10**

votes

**2**answers

681 views

### Categorification of coends and ends

I describe below a categorified version of the coend construction, "2-coend" for short. It takes as input a collection of 1-categories $\{W_{xy}\}$ which afford left and right representations of a (...

**4**

votes

**0**answers

237 views

### Categorical formalism for higher non-abelian group cohomology / obstruction theory for gerbes?

I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question:
What's the neat abstract framework for obstruction theory for non-abelian ...

**11**

votes

**2**answers

433 views

### What are the reflective subcategories of the category of presentable categories?

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well.
A nice property of presentable $\infty$-categories is that if ...

**11**

votes

**2**answers

749 views

### Relation between fully-extended TQFT and a “topless” TQFT

Consider 3-dimensional TQFTs for example. One version of them is the
3-2-1-0 fully extended TQFT. Do we have another version: 2-1-0 extended "TQFT"?
If yes, do we have an example of 2-1-0 extended ...

**10**

votes

**2**answers

3k views

### Is there a meaningful difference between biased and unbiased composition?

In higher category theory, there are notions of biased and unbiased definitions of composition of $n$-morphisms (or, as a special case, tensor products of objects). In the biased framework, we define ...

**9**

votes

**2**answers

564 views

### Is the site of (smooth) manifolds hypercomplete?

By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...

**6**

votes

**4**answers

1k views

### What is the intuition of connections for cubical sets?

I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, ...

**4**

votes

**1**answer

697 views

### references for models of stable infinity categories

There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...

**12**

votes

**2**answers

855 views

### $K$-theory backwards

Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed?
That is, ...

**11**

votes

**1**answer

517 views

### Are $(\infty,1)$-categories $A_\infty$ categories?

Let $X$ be a set. One can define a non-symmetric colored version of the non-unital $A_\infty$ operad as follows. The set of colors is the set of ordered pairs in $X$. Let $(x_1,y_1),\dots,(x_n,y_n)$...

**6**

votes

**0**answers

583 views

### Commutation of simplicial homotopy colimits and homotopy products in spaces

Edit: The claim below is wrong, as explained in the comments,
because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy ...

**4**

votes

**4**answers

457 views

### Equalizer completion

Can anybody give a definition of the equalizer completion of a cartesian category?
Is the method to get more or less as the regular and exact completions in the way that are given in: http://ncatlab....

**2**

votes

**1**answer

442 views

### A simple definition of n-category [closed]

Maybe there is no simple definition of $n$-category understandable for a physicist. Then I would like to know what are the trivial $0$-category, trivial $1$-category, trivial $2$-category, etc. How ...

**15**

votes

**1**answer

651 views

### The state of the art in the rectification of homotopy-coherent structures

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of:
Cordier and Porter proved a ...

**9**

votes

**1**answer

357 views

### Adjoining adjoints in a 2-category

For a given 2-category $C$, does there exist a faithful and locally faithful 2-functor $C \to C^*$, such that the image of every 1-morphism of $C$ has a right adjoint in $C^*$?
Below are some of my ...

**9**

votes

**1**answer

400 views

### Local smallness and (higher) topoi

The $2$-category of topoi and geometric morphisms is not locally small. For example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is ...

**6**

votes

**2**answers

267 views

### Homotopy function complex for quasi-categories

The model structure on sSet for quasi-categories (the Joyal model structure) is not enriched over sSet with the Quillen model structure, so ordinary internal Hom of simplicial sets is not a correct ...

**5**

votes

**2**answers

188 views

### Does the Day convolution induce the structure of a bimonoidal category on $Fun(C,D)$?

Let $\mathcal{C}$ and $\mathcal{D}$ be symmetric monoidal categories and assume that the symmetric monoidal product $\otimes_{\mathcal{D}}$ on $\mathcal{D}$ preserves colimits in both variables.
Then ...

**5**

votes

**1**answer

386 views

### Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?

Let $C$ be an $(\infty,1)$-topos. The $(\infty,1)$-category of group objects in $C$ is a full sub-$(\infty,1)$-category of groupoid objects in $C$:
$${\mathsf{Grp}}(C) \hookrightarrow {\mathsf{Grpd}}(...