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2
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0answers
246 views

Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry. Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...
7
votes
1answer
266 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
3
votes
0answers
177 views

(∞,n)-category of triangulated cobordisms

What is an accepted definition of a (∞,n)-category of triangulated cobordisms? Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
16
votes
2answers
475 views

Homotopy-theoretic derived Morita equivalences

Recall that two $k$-algebras $A, B$ are Morita equivalent iff their categories of left modules are equivalent. However, this relation turns out to be rather fine and one introduces a coarser ...
5
votes
2answers
278 views

How does one Segal-subdivide a 2-category?

Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...
3
votes
2answers
220 views

Localisation in a quasi-category

Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction. Now consider ...
33
votes
4answers
2k views

Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
16
votes
1answer
2k views

Is Lemma A.1.5.7 in Higher Topos Theory correct?

Hello to everyone, I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in section A.2.6. At some ...
2
votes
2answers
364 views

how to make the category of chain complexes into an $(\infty,1)$-category

Related to this question, I would like to know, if there is an explicit presentation of the $(\infty,1)$-category of a model category of abelian chain complexes, but this time in terms of simplicial ...
9
votes
0answers
163 views

KK-theory by abelianized correspondences of smooth stacks?

Whith (Kasparov, bivariant) KK-theory I am left with the nagging feeling that the theory is "more fundamental", than has been made explicit, that there is a "more profound" universal characterization ...
5
votes
2answers
311 views

Is the site of (smooth) manifolds hypercomplete?

By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
6
votes
1answer
393 views

Diagram spectra and Algebraic Geometry

I was recently reading a paper titled "Model Categories of Diagram Spectra" and it was mentioned in the paper that the contents of the paper were also useful in algebraic geometry. I'm really ...
12
votes
0answers
428 views

What are some examples of weak ω-categories?

As is usual, let's say an (n, k)-category is something with objects, morphisms, 2-morphisms, ..., n-morphisms, such that all j-morphisms for j > k are invertible, everything meant in the weak sense. ...
6
votes
3answers
1k views

Infinity-categories vs Kan complexes

Hi all, It is known (cf. Lurie's book "Higher Topos Theory", for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan ...
10
votes
2answers
374 views

What is the Q-construction, metaphysically?

An exact (small) category $P$ is an environment in which we make sense of the "put-together"-edness of objects via (short) exact sequences. It seems like the K-theory of an exact category encodes the ...
4
votes
0answers
205 views

looping and delooping spaces and categories

I'm trying to understand the relationship between the notions of looping and delooping in category theory and topology. The morphisms in a category with one object have the structure of a monoid. ...
4
votes
1answer
294 views

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 ...
5
votes
2answers
1k views

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 ...
2
votes
0answers
78 views

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: ...
8
votes
0answers
354 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 ...
9
votes
1answer
722 views

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 ...
4
votes
1answer
437 views

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 ...
8
votes
1answer
597 views

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 ...
6
votes
1answer
259 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 ...
6
votes
0answers
109 views

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 ...
7
votes
1answer
242 views

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 ...
20
votes
4answers
1k views

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, ...
13
votes
1answer
414 views

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" ...
4
votes
1answer
250 views

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 ...
25
votes
1answer
748 views

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 ...
9
votes
1answer
339 views

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 ...
11
votes
1answer
306 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 ...
15
votes
2answers
722 views

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 ...
8
votes
2answers
348 views

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 ...
1
vote
1answer
544 views

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 ...
8
votes
2answers
738 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 ...
2
votes
2answers
427 views

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 ...
2
votes
0answers
89 views

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 ...
2
votes
2answers
200 views

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 ...
8
votes
0answers
432 views

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, ...
4
votes
0answers
259 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 ...
10
votes
1answer
856 views

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
1answer
281 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 $F\dashv G$ ...
1
vote
2answers
378 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 ...
7
votes
2answers
398 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 ...
11
votes
2answers
457 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 ...
2
votes
2answers
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 ...
3
votes
1answer
171 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. ...
7
votes
0answers
423 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 ...
7
votes
1answer
313 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 ...