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

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### Slices of infinity sheaves

I know from classical category theory that if $C$ is a small category and $X$ is a presheaf, that there is a canonical equivalence $$Set^{C^{op}}/X \simeq Set^{\left(C/X\right)^{op}},$$ where $C/X$ is ...

**3**

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**1**answer

312 views

### Can we exhibit the 2-category of Grothendieck fibrations as a 2 (or 3)-limit?

It's well known that we can exhibit the comma category as a particular type of 2-limit in Cat. When working with 2-categories, there is a naïve comma object given by by boosting up the ordinary ...

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**2**answers

531 views

### Historical Question about Simplicial Sets

I have a pretty easy historical question about simplicial sets. Unless I am mistaken, simplicial sets first came out of topology, explicitly from combinatorial topology and the study of simplicial ...

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**1**answer

438 views

### Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?

If you take the nerve of a groupoid, you get a Kan complex.
Question:
Take a bicategory that has adjoints for 1-morphisms, which is one notion of 'weak' groupoid (if all 2-morphisms are ...

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**3**answers

940 views

### Complicating an Example by Toen (motivations for DAG)

I'm trying to read (the introduction of) a survey by Toen on Derived Algebraic Geometry, specifically the "Simplicial Presheaves and Derived Algebraic Geometry" one.
He motivates the introduction of ...

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**0**answers

292 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 ...

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**2**answers

631 views

### Are the axioms for higher category-theory effectively computable?

I ask this, although I don't conduct any research in the area, or even plan to. -- There seems to be general agreement that the axioms for higher categories grow very rapidly in complexity as the ...

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581 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 ...

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**1**answer

490 views

### co-$A_\infty$ spaces

A co-$A_n$ space is a based space $Y$ equipped with a co-action by the Stasheff associahedron operad $K_\bullet$. This means that $Y$ is comes with certain maps $c_n: Y \times K_n \to Y^{\vee n}$, $n ...

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**4**answers

379 views

### What properties should a good definition of (weak) $n$-category satisfy?

My (perhaps inaccurate) impression is there are many competing definitions of the notion of a (weak) $n$-category, none of which are generally accepted. I've run across several properties such ...

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**1**answer

437 views

### Is there a good notion of “induction” for representations of 2-categories?

One of the most important observations in the representation theory of algebras is that if one has a subalgebra $A\subset B$, then these is a functor $B\otimes_A -\colon A\operatorname{-pmod}\to ...

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**1**answer

744 views

### Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory

Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT ...

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**1**answer

272 views

### The plus construction for stacks of n-types

In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+1\right)$ times, and in general, ...

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360 views

### $\infty-$groupoid of $A_{\infty}$ algebras

Hello,
Consider first the following $2-$groupoid of Algebras over $\mathbb{C}$. Objects are Algebras, $1-$morphisms are isomorphisms, and a $2-$morphism between the isoms $f$ and $g$ from $A$ to $B$ ...

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**2**answers

313 views

### Do quasi-categories have a `completion'?

So I just started learning about quasicategories... Alright, that's an understatement: I just listened to Julie Bergner talk about quasicategories, and then started reading Moritz Groth's short course ...

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**0**answers

401 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 ...

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**2**answers

607 views

### Connections in double categories

There exist a structure on double categories due to R.Brown called a connection. The connection embodies in squares an isomorphism between the category of its vertical arrows and the category of its ...

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**1**answer

249 views

### Lax universality for lax limits

A lax limit is defined to be a 2-limit, except that the cone is only required to commute up to specified transformations, not up to isomorphism. In particular, the limit is defined up to isomorphism, ...

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**3**answers

225 views

### Are fibrations coreflective in a 2-category?

The notion of a Grothendieck fibration in the 2-category $CAT$ of small categories, can be written down to make sense for any 2-category, and such a morphism in a 2-category is called a fibration:
...

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**1**answer

384 views

### Colimits of topological groupoids

Let $G$ and $H$ be two topological groupoids. Suppose that I have two morphisms $G \rightrightarrows H$ and I want to take the 2-coequalizer of these maps. I'd like an explicit description of (a ...

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**1**answer

301 views

### Weak colimits of weak and strict presheaves in groupoids

Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and ...

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**4**answers

312 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: ...

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**3**answers

476 views

### Chain Complexes and Linear Infinity-Categories

A statement I heard recently is that "chain complexes are the same thing as strict linear $\infty$-categories". Can someone explain how to see this?

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**4**answers

729 views

### Homotopic quotients of simplicial sets as infinity-groupoids

Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of ...

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### The space of framed functions

Framed functions arose in the work of K. Igusa defining cohomology invariants for smooth manifold bundles (Igusa-Klein torsion). In the late 80's, he proved a strong connectivity result about the ...

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**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 ...

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**3**answers

505 views

### Is there a tricategory of bicategories and biprofunctors?

Background
There is a bicategory where the objects are categories, the 1-morphisms are profunctors, and the 2-morphisms are morphisms of profunctors. The non-obvious part of this assertion is that ...

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votes

**2**answers

278 views

### 2-groups are to crossed modules as 2-categories are to…?

Given a 2-group $\mathcal{G}$, you can construct a crossed module $(G,H,t,\alpha)$ and vice versa.
Is there something similar you can say for strict 2-categories?
In a personal attempt to understand ...

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**1**answer

325 views

### Is a pseudonatural transformation of strict 2-functors to Cat isomorphic to a 2-natural transformation?

Let $\mathcal{C}$ be a strict 2-category. A corollary of the bicategorical Yoneda lemma says that any pseudofunctor $\mathcal{C} \to \operatorname{Cat}$ is pseudonaturally equivalent to a strict ...

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338 views

### nerves of crossed complexes, group T-complexes and classifying spaces

A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.
There are a couple of ...

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659 views

### What makes the stable module category stable?

When geometrically flavoured words like "mapping cone" or "chain homotopy" crop up in homological algebra, there's usually a good reason. (In this case, looking at the chain complexes associated with ...

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**1**answer

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### What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...

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### What's the current state of the classification of not-fully-extended TQFTs?

Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some ...

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**1**answer

889 views

### t-structures and higher categories?

I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer).
Given a triangulated category, one ...

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### 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 ...

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**1**answer

181 views

### $\omega$-monoids

Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of ...

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**1**answer

737 views

### A Model Category of Segal Spaces?

So in Julie Bergner's work on (infty, 1)-categories arXiv:0610239, she considers several model categories which model (infty, 1)-categories, which are known to be equivalent. I'm guessing that there ...

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279 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 ...

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**1**answer

552 views

### When is a stack (NOT) geometric?

Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ ...

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**1**answer

664 views

### Can equivalences be strictified to isomorphisms?

In category theory there are lots of examples of isomorphisms that cannot be strictified to become identities. For instance, every monoidal category is equivalent to a strict monoidal category, where ...

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**1**answer

376 views

### Double Category of Topological Stacks

There are two equivalent ways of describing topological stacks.
One is the "stacky" definition, that is, a topological stack is a stack $\mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if ...

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**0**answers

325 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 ...

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**4**answers

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### Cotangent bundle of a differentiable stack

If you ever wanted to construct the tangent bundle of a differentiable stack, it's relatively simple:
First, if $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$, you could just say ...

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**6**answers

3k views

### 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 ...

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378 views

### (∞,1) vs Category weakly enriched over spaces

What is the difference between:
($\infty,1$) categories - in which have for two objects you have an ($\infty,0$) category of morphisms (i.e. a space of morphisms)
and
categories weakly enriched ...

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**1**answer

336 views

### Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to \left\[G\right]$, where $\left\[G\right]$ is the ...

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**1**answer

303 views

### Monoidal closed structure(s) on the category “bicategories, with strict functors”?

I'm working with globular operadic higher categories (with the Batanin/Leinster definitions) and ending up working a lot in the categories $P$-$\mathrm{Alg}$ of algebras for some globular operad $P$, ...

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### What functor is adjoint to the tensor product of 2-vector spaces?

This is a refinement of my (naive, poorly asked) question here. The reference for my question is Baez and Crans, HDA6.
Background: category objects, etc.
Let $\mathcal V$ be a category. A category ...

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708 views

### A question on group action on categories

Let $Gr$ be the affine Grassmannian of $G=G((t))/G[[t]]$, and let $Perv(Gr)$ be the category of perverse sheaves on $Gr$. We have action of $G((t))$ on the left-hand side of $Perv(Gr)$, also we have ...

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### Lax and Colax Monads

Is there much known about the theory of lax and colax monads on a bicategory? Here, I really mean lax or colax, not weak. I'm aware of some literature about weak monads. I'm interested in distributive ...