# Questions tagged [higher-category-theory]

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

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### Is every left fibration of simplicial sets with nonempty fibers a trivial kan fibration?

In Lemma 2.1.3.4 of Higher Topos Theory, the statement of the lemma requires that the fibers are not only nonempty but contractible. However, in the proof, I don't see where contractibility is ...

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### Models for, and motivation for, (oo,n)-categories for general n

First: Is there a precise meaning to the term "model for (oo,n)-categories"? A related question, which might actually be the same question, is: what exactly do we want to get out of (oo,n)-category ...

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### 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 ...

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### categorification of logic

has there be an effort to categorify first order logic? More particularly, structures in the sense of logic.
If so, then every structure of a first order theory is a category. so in particular, the ...

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**2**answers

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### What are natural transformations in 1-categories?

It's well-known that, for lots of concrete categories (but by no means all), we can think of the objects as themselves being small categories, and morphisms are the functors between these categories. ...

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### (infinity,1)-categories directly from model categories

Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" ...

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### Simplicial homotopy book suggestion for HTT computations

I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble I'...

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**1**answer

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### Generation of object-internal structure in a strict 2-category

Suppose we're given a strict and small 2-category $C$, and an object of $C$ called $A$. Can we produce an internal category structure on $A$ in some canonical way (maybe by some sort of argument ...

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### 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 ...

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### Are bicategories of lax functors also bicategories of of pseudofunctors?

Let A be a bicategory. Can I construct in some "natural" way a bicategory L(A) such that for every bicategory B, the bicategory of lax functors Lax(A, B) is (bi)equivalent to the bicategory of ...

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### Lax Functors and Equivalence of Bicategories?

Lax functors of bicategories were introduced at the very inception of bicategories, and I'm trying to get a better feel for them. They are the same as ordinary 2-functors, but you only require the ...

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### 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 ...

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### 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 ...

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### 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**

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**1**answer

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### (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**

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**1**answer

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### 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 ...

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**1**answer

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### 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 ...

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### $\omega$-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 ...

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### 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**

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**1**answer

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### 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 ...

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### 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 ...

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### 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 ...

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**1**answer

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### 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 $n \geq 3$? ...

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**1**answer

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### 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, ...

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### 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 ...

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### 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 ...