# Questions tagged [higher-category-theory]

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

735
questions

**4**

votes

**3**answers

869 views

### Computation of Joins of Simplicial Sets

It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a boundary?, etc? and ...

**7**

votes

**3**answers

789 views

### Joins of simplicial sets

Why doesn't the join operation on the category of simplicial sets commute up to unique isomorphism? I mean, aren't products and coproducts commutative up to isomorphism? That leads me to conclude at ...

**8**

votes

**2**answers

709 views

### Higher order quandle

The notion of quandle is known to be closely related to knot theory. The three axioms in the definition of quandle correspond to the Reidemeister moves.
Recently I learned that there are higher ...

**3**

votes

**1**answer

370 views

### (n+1,r+1)-Theta space of (n,r)-Theta spaces?

I started writing nLab:Theta space. Not done yet, but while I am working on it:
is there a good proposal for what the "$(n+1,r+1)$-$\Theta$-space of all $(n,r)$-$\Theta$-spaces" would be?

**5**

votes

**1**answer

587 views

### Local Joyal-simplicial presheaves?

It is well known that left Bousfield localizations of the global functor model category $Func(C^{op}, SSet_{standard})$ of functors with values in simplicial sets equipped with the standard model ...

**6**

votes

**1**answer

319 views

### Relation between dendroidal and opetopic sets

To my shame I have to admit that I have as yet not looked much into opetopes and opetopic sets.
I am in the process of writing nLab entries on dendroidal sets and noticed that some remarks on the ...

**10**

votes

**3**answers

1k views

### $\omega$-topos theory?

I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$-toposes, which leads me to the following question: Is there any sort of higher topos theory ...

**4**

votes

**2**answers

202 views

### What is a cograph of an n-functor?

I'm trying to get my head around what a cograph of an n-functor is. We (some n-Lab people) are discussing it here. As a start, I'd be happy to understand what the cograph of a 0-functor, i.e. function ...

**5**

votes

**1**answer

379 views

### Where does the “easy” definition of a weak n-category fail?

Okay, I'm going to ask a naiive question that surely has an interesting answer. So, a first approximation of defining a (small) weak n-category probably goes something like this. Take a pre-n-category ...

**8**

votes

**4**answers

582 views

### What's the right object to categorify a braided tensor category?

The yoga of categorification has gained a lot of popularity in recent years, and some techniques for it have made a lot of progress. It's well-understood that, for example, a ring is probably ...

**11**

votes

**2**answers

524 views

### What do decategorification and “compactification on a circle” have to do with each other?

Some physicists have told me that if you think about an extended n-dimensional TQFT $F$, then the decategorification is given by $F'(X)=F(X\times S^1)$, which I believe they call "compactification on ...

**3**

votes

**1**answer

343 views

### omega-categories and n-fold complete segal spaces

Why are $n$-fold complete segal spaces or $(\infty, n)$-categories (which I'm unsure of how to distinguish from omega-categories) important for $n \geq 3$? Why are they "badly behaved" for $n \geq 3$? ...

**4**

votes

**1**answer

561 views

### Eckmann-Hilton argument

The Eckmann-Hilton argument is used to prove that a doubly monoidal 0-category is a commutative monoid. If (x) is horizontal composition and . is vertical composition, and assuming that 1(x)a=a=a(x)1, ...

**8**

votes

**4**answers

2k views

### Prestacks and fibered categories

It seems to be a well-known fact that there is a "one-to-one correspondence'' between prestacks and fibered categories. Here a prestack (called a pseudo-functor in SGA1) means a contravariant lax ...

**16**

votes

**2**answers

1k views

### What's an example of an “adjunction up to adjunction”?

(There are several dialects of 2-categorical language. Mine is the one where everything is weak by default. You might need to insert the word "weak" or "pseudo" (or even "strong", if your default is ...