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8
votes
2answers
773 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
431 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
203 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
446 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
265 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
882 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
297 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
383 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
405 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
463 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
301 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
183 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
426 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
323 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
280 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
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 ...
6
votes
4answers
697 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
319 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 ...
21
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
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 ...
7
votes
1answer
565 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
214 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 ...
8
votes
0answers
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 ...
4
votes
2answers
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? ...
5
votes
1answer
310 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
193 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
870 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
542 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
446 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
582 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
575 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
538 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
182 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
322 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 ...
1
vote
1answer
176 views

gluing bundles as a 2-colimit

Is the gluing of bundles from not-necessarily trivial bundles just some kind of 2-colimit?
2
votes
3answers
999 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 ...
3
votes
1answer
251 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 ...
8
votes
4answers
640 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 ...
6
votes
1answer
338 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 ...
7
votes
3answers
595 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 ...
6
votes
3answers
709 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 ...
4
votes
1answer
302 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 ...
6
votes
1answer
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 ...
9
votes
0answers
510 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). ...
5
votes
2answers
290 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 ...