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

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### What is a higher derived constructible sheaf

Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C_*(\Omega X,k)$ of ...

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### Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...

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### Bisections in Kan Complexes

Kan Complexes can be seen as a generalization of groupoids, mostly called (weak)
infinity groupoids in this context.
On groupoids we can define the \textbf{group of bisections} the following way:
...

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347 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 ...

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### Are there non-categorical notions in topos theory?

Suppose that $\mathcal{T}$ is an abstract $2$-category we know is equivalent to the $2$-category of Grothendieck topoi via some equivalence $$\phi:\mathcal{T} \to \mathfrak{Top},$$ and let $E$ be an ...

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### Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?

This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...

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**1**answer

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### Are Thom spectra MU, MSO and K-theory spectra KU, KO modules over some truncations of the sphere spectrum?

The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO).
On the other hand, MU and ...

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255 views

### When is this braiding not a symmetry?

Given a topological space $X$ instead of forming the fundamental groupoid $\pi(X)$ which is the category whose objects are the points and morphisms the homotopy classes of paths one can also form the ...

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### 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 ...

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### Analogue of cyclic homology for e_n-algebras?

Cyclic homology may be defined as the primitive part (with respect to a natural product) of the homology of the Lie algebra associated with the "stabilization" of an associative algebra $A$. Here the ...

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### What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...

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### Skeleton category of the category of skeleton categories?

A category is a skeleton if, roughly speaking, no two distinct objects within the category are isomorphic. To every category is associated a skeleton, and two categories are categorically "equivalent" ...

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### Is a conservative finite limit preserving functor of (infinity,1)-categories homotopically faithful?

In classical category theory we have a following criterion. If $\mathcal{C}$ and $\mathcal{D}$ are finitely complete categories and $F : \mathcal{C} \to \mathcal{D}$ is a functor which preserves ...

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### n-categorical description of Chern classes

The Chern classes of a rank $n$ vector bundle on $X$ are obtained from composing the associated classifying map in $[X, BU(n)]$ with the maps $BU(n) \to B^{2i} \mathbb{Z}$ corresponding to the ...

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### Model for the (infinity,1)-category of functors preserving certain homotopy limits

This question is a follow up to: Model for the (infinity,1)-category of (homotopy-)limit preserving functors.
Warm-up Question: Given a simplicial model category $M$, what model category models ...

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302 views

### Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...

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### Internal categories in simplicial sets

Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories?
Note that this ...

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### Equivalences of Internal Categories

I am trying to understand double categories and their relatives a little better. I do not understand much, so I apologize in advance if my question is too naive for this website.
A functor $F:X ...

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**1**answer

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### Where should one go to learn about triangulated categories?

Lurie's book, higher topos theory describes a new notion of a triangulated category, which is apparently much more natural than the usual definition. Obviously by now a great deal of work has been ...

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715 views

### Are grothendieck universes enough for the foundations of category theory?

Grothendieck universes are equivalent to ZFC+a strongly inaccessible cardinal. This is low on the large cardinal axiom list. Is it enough to place category theory on a firm foundational basis, and how ...

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### Is it possible to approch higher categories from the arrow functor?

I have not read anything throughly on higher categories. I am only consuice that in higher categories, we have higher dimensional cells, after adjusting the intuition that 0, 1, 2 dimensional cells ...

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### What about lax-coherence results?

As well know in category theory (see Mac Lane book) there are (very useful, and theoretically deep) coherent theorem for various categorical setting: monoidal, symmetric monoidal, braided monoidal ...

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### A (too easy) normalization of a lax-funtor between 2-categories ?

Let $(F, \phi): \mathscr{A}\to \mathscr{B}$ a lax-functor between 2-categories. In the setting of 2-categories the axiom of lax-functor become:
Ul) $1: F(f)=F(f)\circ 1_{F(X)}\xrightarrow{1\circ ...

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### 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, ...

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257 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 ...

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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, ...

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### [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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### 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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390 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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### 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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### 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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**1**answer

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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**1**answer

219 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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### 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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208 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...