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

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### How much do universes matter in topos theory?

Suppose we fix two Grothendieck universes $\mathcal{U} \in \mathcal{V}.$ Then one has that $\mathcal{U}$-$\mathbf{Set},$ the category of $\mathcal{U}$-small sets, is a locally $\mathcal{U}$-small, ...

**6**

votes

**1**answer

294 views

### [Reference Request] The Definition of Adjoint Functors between dg-categories

Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair?
In my mind $F\dashv G$ ...

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vote

**2**answers

381 views

### Internal Homs in Infinity Categories

Given an $\infty$-category $\mathcal{C}$ with whatever descriptors you wish, how do higher internal homs work? Specifically, given two $n$-morphisms, $f$ and $g$, is it possible (I don't see why not a ...

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401 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 ...

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461 views

### What are the higher morphisms between enriched higher categories?

This question is about $n$-categories, or perhaps $(\infty,n)$-categories, or ... My guess is that the answer will not depend sensitively on the model of higher categories, so rather than have me ...

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**2**answers

300 views

### functors unique up to self-equivalence of the source category

Call two functors two functors $H,H':S\longrightarrow T$ weakly equivalent, or equivalent up to a self-equivalence of the source category, iff
there exists a self-equivalence of $s:S \longrightarrow ...

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votes

**1**answer

181 views

### Need M combinatorial for existence of injective model structure on $M^G$?

I'm doing some work with model categories and operads, and to check a certain hypothesis I've had to learn a bit of equivariant homotopy theory. Let $M$ be a model category and $G$ be a finite group. ...

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425 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 ...

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votes

**1**answer

319 views

### String diagrams for (weak) monoidal categories

Hi,
In a strict monoidal category, where the associator, left and right unitor are identity morphisms we have the following relations between (string) diagrams:
where $i_x$ and $e_x$ are the unit ...

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277 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 ...

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**1**answer

381 views

### n-categories enriched in (n+1)-categories

Recall the notion of an $n$-cateogry $C$ enriched in a symmetric monoidal category. Instead of a set of $n$-morphisms $mor(a, b)$ (where $a$ and $b$ are compatible $(n{-}1)$-morphisms), we have an ...

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**4**answers

694 views

### Is there a sense in which the homotopy theory of simplicial sets is the “paradigmatic” one?

I could not come up with a better title for my question.
What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):
the ...

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votes

**1**answer

318 views

### Are reflective subcategories of complete infinity categories complete?

It is well known that reflective subcategories of complete categories are complete, and that limits in the subcategory are computed by taking the limit in the ambient category and applying the ...

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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 ...

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**3**answers

1k views

### Higher categories as data structures

Still wading through higher category theory. I find the subject a bit intimidating, not so much for technical reasons, but because I lack sufficient intuition as to the motivation(s)/heuristics one ...

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votes

**1**answer

225 views

### Is there a name for this type of equivalence in a 2-category?

Recall in a $2$-category $X$, a $1$-cell $f:X\to Y$ is called an equivalence provided there exists a $1$-cell $g:Y\to X$ together with the data of a pair of isomorphisms $\eta_X: gf \to ...

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557 views

### Untyped Higher Category Theory

I am currently trying to wade through the vast lake of higher category theory, a formidable task,or so it seems.
In the process, it has occurred to me that there is a basic analogy in place with ...

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vote

**1**answer

210 views

### Do Cartesian fibrations preserve pullbacks?

If $F:C \to D$ is a Cartesian fibration of $\infty$-categories, I would like to show that $F$ preserves pullbacks. This seems intuitively clear, but I haven't found it in HTT (but perhaps I missed ...

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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 ...

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**2**answers

252 views

### Is there a notion of “ribbon 2-category”?

It there some notion of ribbon 2-category, which would allow for, say, talking about the Seifert surface of links (which is a 1-morphism in some ribbon category) as a 2-morphism in the category?
...

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**1**answer

308 views

### A slicker proof that an object must be initial

If $\mathcal{C}$ is a category and $\lambda:\Delta_D \to id_{\mathcal{C}}$ is a cone for the identity functor, and $F:J \to \mathcal{C}$ is a functor such that $F\lambda:\Delta_D \to F$ is a limiting ...

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**0**answers

190 views

### On small generators of an infinity category

Suppose that $\mathcal{C}$ is an $\infty$-category with pullbacks and small coproducts. Suppose that $\mathcal{D}$ is a small subcategory for every object $C,$ the canonical map ...

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**1**answer

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### Pseudofunctors out of the lax Gray tensor product

I feel like I should know the answer to this, but I don't think I do.
The Gray tensor product of 2-categories $C$ and $D$ is a "fattening up" of the cartesian product $C\times D$ in which the ...

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vote

**1**answer

315 views

### Descent of Morphisms of Sheaves

While reading Brylinski I am trying to understand the descent of morphisms of sheaves.
In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local ...

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540 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 ...

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437 views

### Canonical topology for infinity topoi revisited.

A while ago I asked this quetion: Canonical topology for big infinity topoi
and this question: How to resolve size issues with the regular epimorphism topology
Let me first summarize some of what I ...

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**1**answer

580 views

### compact objects in model categories and $(\infty,1)$-categories

In an ordinary category $C$, one says that an object $X$ is $\kappa$-compact if the representable functor $Hom(X,-)\colon C \to Set$ preserves $\kappa$-filtered colimits. We say $C$ is locally ...

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**1**answer

246 views

### Pushforwards of stacks of algebras?

This is a refined/sheafified version of this previos question of mine.
Let $(X,\mathcal{O}_X)$ be a ringed space or more in general a ringed stack, where the structure sheaf $\mathcal{O}_X$ is a ...

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**1**answer

568 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 ...

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### 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 ...

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**1**answer

181 views

### local model structure on simplicial presheaves

Hello,
Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.
Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its ...

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278 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 ...

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**2**answers

317 views

### Cartesian cubes and groupoids

Given a groupoid $G,$ one can consider the canonical epimorphism $$G_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram ...

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173 views

### gluing bundles as a 2-colimit

Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?

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**3**answers

971 views

### Exact sequences in homotopy categories

I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology.
How ...

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**1**answer

248 views

### How can I see that the slice of a presheaf category is equivalent to the presheaf category of the category of elements?

Let $\mathcal{C}$ be a small category and $P$ a presheaf on $\mathcal{C}$. Then there is an equivalence $$\widehat{\mathcal{C}} / P \cong \widehat{\int_{\mathcal{C}}} P$$ Now there are (at least) two ...

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**4**answers

636 views

### Higher categories in logic

I've read somewhere (probably in the nlab) that higher category theory has application in logic.
By the way since now the only applications of higher category theory I've seen are in homotopy theory ...

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**1**answer

335 views

### Detecting equivalences of (infinity) categories by nerves

I have two questions:
Is there a way to tell if a functor $F:C \to D$ between two small categories is an equivalence in terms of the map $$N(F):N(C) \to N(D)$$ between simplicial sets? More ...

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590 views

### A 2-category of chain complexes, chain maps, and chain homotopies?

First-time here... I hope my question isn't silly or anything... anyway...
Consider the category of chain complexes and chain maps. We can also define chain homotopies between chain maps. Does this ...

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**3**answers

703 views

### Segal's Original Definition of a Topological Category

Nowadays we can associate to a topological space $X$ a category called the fundamental (or Poincare) $\infty$-groupoid given by taking $Sing(X)$.
There are many different categories that one can ...

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votes

**1**answer

300 views

### Writing an infinity groupoid as a colimit of sets

If we are given a simplicial set $X:\Delta^{op} \to Set$, we may regard it as a \emph{simplicial} simplicial set, i.e. a bisimplicial set by composing with the "constant" inclusion $Set \to ...

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352 views

### Why are sheaves not preserved in this case?

Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi_{\mathscr{X}}:\int_{C} \mathscr{X}\to C$$ denote the associated fibered ...

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508 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).
...

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288 views

### Fibration of Batanin/Leinster $\omega$-groupoids

Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster?
I tried to search on Google and in Higher Operads, Higher Categories of Tom ...

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**1**answer

432 views

### How do various notions of natural transformation relate to various notions of homotopy in $2Cat$?

In what follows, $2$-categories will be strict, and "$2$-functor" will mean "strict $2$-functor". (Please mention which terminological conventions you are using when answering.) I guess that the ...

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### Why is the Street nerve of the Gray tensor product $[1]\otimes [1]$ isomorphic to $[1]\times [1]$

Recall that given two strict ω-categories $A$ and $B$, their lax Gray tensor product $A\otimes B$ is sent to the Verity-Gray tensor product of their associated complicial sets ...

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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 ...

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### 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 ...

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### 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 ...

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### Weak algebraic structures

The following question can be thought as a sequel of this one.
Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...