The higher-category-theory tag has no wiki summary.

**21**

votes

**0**answers

653 views

### categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...

**12**

votes

**0**answers

434 views

### What are some examples of weak ω-categories?

As is usual, let's say an (n, k)-category is something with
objects, morphisms, 2-morphisms, ..., n-morphisms, such that all
j-morphisms for j > k are invertible, everything meant in the
weak sense. ...

**11**

votes

**0**answers

346 views

### Chiral categories versus braided monoidal categories

Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of ...

**11**

votes

**0**answers

211 views

### Is there a quantum group or loop group description of a braided monoidal 2-category giving Khovanov homology?

Recall that there are (at least) two ways to describe the modular tensor category that $3$-dimensional Chern-Simons (with gauge group $G$ and level $k$) assigns to a circle: one involving ...

**11**

votes

**0**answers

617 views

### Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...

**10**

votes

**0**answers

280 views

### What is the history of the notion of subdivision of categories?

A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...

**9**

votes

**0**answers

176 views

### “extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...

**9**

votes

**0**answers

172 views

### KK-theory by abelianized correspondences of smooth stacks?

Whith (Kasparov, bivariant) KK-theory I am left with the nagging feeling that the theory is "more fundamental", than has been made explicit, that there is a "more profound" universal characterization ...

**9**

votes

**0**answers

510 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

**0**answers

438 views

### Is there any elementary text unravelling the definitions of 2-category, lax functor and lax transformation, allowing people who do not know in the first place what these things are to really understand the definitions?

The question is in the title.
My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people ...

**8**

votes

**0**answers

191 views

### Are n-truncated quasicategories a model for n-categories?

In Higher Topos Theory, Lurie gives a description of $n$-categories (section 2.3.4) in terms of quasicategories. In other words, he gives a definition (2.3.4.1) of an $\infty$-categorical notion of ...

**8**

votes

**0**answers

369 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 ...

**8**

votes

**0**answers

446 views

### Compact objects in triangulated and infinity categories

Hello,
I think of a compact object in a category as an object, $Hom$ from which commutes with filtered colimits.
I guess that in an infinity category, one also defines a compact object as an object, ...

**8**

votes

**0**answers

248 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 ...

**8**

votes

**0**answers

432 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 ...

**8**

votes

**0**answers

300 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 ...

**8**

votes

**0**answers

415 views

### E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product

Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 ...

**7**

votes

**0**answers

78 views

### What suffices to check completeness in an n-fold Segal space?

Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the
Segal condition: For each $j$, the map
$$ X_j \to ...

**7**

votes

**0**answers

190 views

### Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...

**7**

votes

**0**answers

426 views

### Where else has Proposition B1.3.17 in the Elephant been proved?

(I asked the same question here and got some helpful comments, but thought I'd re-ask in case I get a more direct response.)
This is a sort of reference request. Proposition B1.3.17 in Johnstone's ...

**7**

votes

**0**answers

538 views

### TQFTs with target category of higher type than the source

In the classical version of the Cobordism Hypothesis, such as, e.g., in Jacob Lurie's On the Classification of Topological Field Theories, one considers the $\infty$-category of symmetric monoidal ...

**7**

votes

**0**answers

280 views

### Generating acyclic cofibrations for the various model structures in higher category theory

There are a number of model categories important in higher category theory, which provide a "presentation" of some $\infty$-category of $\infty$-categories. For example:
The Joyal model structure on ...

**7**

votes

**0**answers

182 views

### Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...

**7**

votes

**0**answers

247 views

### Completion of n-fold Segal spaces

During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for ...

**7**

votes

**0**answers

589 views

### An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point:
Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with
the contravariant model structure (cofibrations are monomorphisms ...

**6**

votes

**0**answers

120 views

### When does a sheaf of categories represent a homotopy sheaf?

Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...

**6**

votes

**0**answers

163 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 ...

**6**

votes

**0**answers

111 views

### Is hypercompletion functorial?

Given an infinity topos $\mathcal{E}$, one can pass to its hypercompletion $\widehat{\mathcal{E}}$, which is the full subcategory on those objects local with respect to $\infty$-connected morphisms ...

**6**

votes

**0**answers

175 views

### Canonical topology for big infinity topoi

It is well known that for $E$ a Grothendieck topos, (by appropriately making use of universes) $E$ carries the canonical Grothendieck topology generated by jointly surjective epimorphisms, say $J$, ...

**6**

votes

**0**answers

192 views

### Good Internal Hom for Weak Complicial Sets?

So I am trying to learn a bit more about Dominic Verity's model of higher categories, namely weak complicial sets. The underlying object is a stratified simplicial set which satisfies a sort of inner ...

**6**

votes

**0**answers

432 views

### Principal $G$-bundles as fully extended TQFTs, and $n$-representations

This is a follow up to this MO question: Fully dualizable objects in classical field theories
Assuming the notation there (which in turn come from Topological Quantum Field Theories from Compact Lie ...

**6**

votes

**0**answers

286 views

### Rigorous recursive definition of $m$-algebras

Naively, $m$-algebras over a field $\mathbb{k}$ are easily defined recursively: the category of 0-algebras is the symmetric monoidal category of $\mathbb{k}$-vector spaces, and for $m>0$ an ...

**5**

votes

**0**answers

135 views

### What is a stable $(n,1)$-category?

My question in its simplest form is whether we have any understanding of stable $(n,1)$-categories, by analogy with stable $(\infty,1)$-categories and abelian categories (the latter being like "stable ...

**5**

votes

**0**answers

150 views

### Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category

Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$.
In HigherAlgebra, the derived category ...

**5**

votes

**0**answers

93 views

### Non-degenerate limits of topoi

Let $\mathcal{E}$ be an elementary topos with natural numbers object $\mathbb{N}$, and let $\mathbb{C}$ be a cofiltered internal category in $\mathcal{E}$. Suppose we have a diagram of shape ...

**5**

votes

**0**answers

290 views

### The span-nerve of a category

Let ${\bf D}$ denote the category of finite non-empty sets and monomorphisms between them; let $\underline{n}$= {$0,1,\ldots,n$}. I will call presheaves on ${\bf D}$ simplicial sets and denote it ...

**5**

votes

**0**answers

328 views

### Chain/Hierarchy of Monoids

Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And ...

**4**

votes

**0**answers

68 views

### Trying to understand straightening functor associated to a right fibration of simplicial sets

Let $p:X \to S$ be a right fibration of simplicial sets; one can roughly think of it as some sort of "functor" $S^{op} \to Set_\Delta$ (where $Set_\Delta$ denotes simplicial sets) sending $s \in S$ ...

**4**

votes

**0**answers

77 views

### Limits in Span(Vec)

Let Vec be the category of real vector spaces and linear maps. Let Span(Vec) be the bicategory of correspondences between real vector spaces. I am trying to understand lax limits in Span(Vec). What ...

**4**

votes

**0**answers

165 views

### $n$-Fold Framed Functions

Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ...

**4**

votes

**0**answers

218 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

**0**answers

83 views

### Limits and colimits in a 2-category vs. in an infinity category: the non- (2,1)-case

Suppose that $\mathscr{C}$ is a $2$-category (or more generally a bicategory) which is not a $\left(2,1\right)$-category. Is there any relation between limits and colimits in $\mathscr{C}$ (in the ...

**4**

votes

**0**answers

128 views

### Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that:
$\mathcal C$ is ...

**4**

votes

**0**answers

219 views

### looping and delooping spaces and categories

I'm trying to understand the relationship between the notions of looping and delooping in category theory and topology.
The morphisms in a category with one object have the structure of a monoid. ...

**4**

votes

**0**answers

265 views

### How to endow an n-fold Segal Space with a symmetric monoidal structure?

I would like to understand how I can endow an n-fold (complete) Segal Space with a symmetric monoidal structure. My question is basically the same as in this post: What is a symmetric monoidal ...

**4**

votes

**0**answers

193 views

### Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

Recall:
Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we ...

**3**

votes

**0**answers

79 views

### Lifting commutative diagrams of functors from the homotopy level to the “higher” level

Let $\mathcal A$ and $\mathcal B$ be differential graded categories over a field. Let $F, G, K : \mathcal A \to \mathcal B$ be quasi-functors (see here for definitions). Assume you have morphisms of ...

**3**

votes

**0**answers

108 views

### When do limits and colimits of infinity-categories commute?

This is a question for someone who read (or wrote) enough of Lurie's HTT to know a reference. Suppose $D,E$ are small "diagram" $(\infty,1)$-categories, and $\mathcal{C}$ is a stable infinity-category ...

**3**

votes

**0**answers

82 views

### How do you compute a homotopy colimit in a category of fibrant objects?

This question may be a bit vague, (so if suggested, can I make it community wiki), but I was wondering what techniques there exists for computing homotopy colimits in a category of fibrant objects. A ...

**3**

votes

**0**answers

113 views

### When does prolongation preserve sheaves?

Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will ...