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6
votes
0answers
112 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
252 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 ...
21
votes
4answers
2k 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
448 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
260 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
763 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
345 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
313 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 ...
16
votes
2answers
756 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
352 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
548 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
796 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
438 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
91 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
204 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
455 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
267 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
895 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
303 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
384 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
418 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
465 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
302 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
184 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
428 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
327 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 ...
10
votes
0answers
285 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 ...
7
votes
1answer
384 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 ...
6
votes
4answers
702 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 ...
7
votes
1answer
321 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 ...
22
votes
2answers
1k views

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 ...
9
votes
3answers
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 ...
2
votes
1answer
226 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 ...
7
votes
1answer
568 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 ...
1
vote
1answer
218 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 ...
9
votes
0answers
254 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 ...
4
votes
2answers
253 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? ...
5
votes
1answer
311 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 ...
2
votes
0answers
195 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 ...
11
votes
1answer
879 views

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 ...
1
vote
1answer
320 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 ...
17
votes
2answers
546 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 ...
7
votes
2answers
468 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 ...
12
votes
1answer
590 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 ...
5
votes
1answer
248 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 ...
6
votes
1answer
579 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 ...
7
votes
0answers
539 views

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 ...
2
votes
1answer
185 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 ...
7
votes
0answers
280 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 ...
3
votes
2answers
323 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 ...