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5
votes
3answers
374 views

What is a Homotopy between $L_\infty$-algebra morphisms II

I would like to continue on question I asking, what is a homotopy between Lie infinity algebras, since I'm not satisfied in two directions: 1.) The naive approach to define a homotopy would be ...
3
votes
2answers
320 views

2 and 3 pullbacks

If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times_C C^I \times_C B$, where $I$ is the ...
3
votes
1answer
254 views

Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology

Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this ...
3
votes
3answers
254 views

Higher dimensional pasting diagram of cubes

A pasting diagram is the analogue of composition of arrows in higher categories. The nlab page gives an example of a two dimensional diagram and how to compute it as a composite of two morphisms. I am ...
3
votes
1answer
160 views

What is the suitable setting for supercoherence with value in a bicategory?

It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as ...
7
votes
1answer
214 views

Are topoi and etale geometric morphisms locally small?

This question is related to this one: Local smallness and (higher) topoi which has not yet been answered. The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in ...
9
votes
2answers
372 views

Do Homotopy Fully Faithful Functors Push-out?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces. Are homotopy fully faithful functors preserved under (homotopy) pushout? More ...
6
votes
1answer
217 views

Joins of, and limits in, $(\infty,1)$-categories via profunctors

I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind. In the classical setting it is almost a triviality to express the join ...
9
votes
1answer
338 views

Local smallness and (higher) topoi

The $2$-category of topoi and geometric morphisms is not locally small. For example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is ...
23
votes
2answers
748 views

How do you define (infinity,1) categories in Homotopy Type Theory?

One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...
2
votes
1answer
156 views

Create 2-category from a cartesian category

I have a way of constructing (something like) a 2-category from a category with products $\mathcal{C}$. My question: Is this a correct construction, and if so, does it have a name? Define ...
7
votes
0answers
190 views

Is there a Rado category?

The Rado graph appears to have a nice universality property (it contains all finite and all countably infinite graphs as induced subgraphs) and homogeinety property (any isomorphism between ...
1
vote
1answer
188 views

Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?

Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
9
votes
1answer
514 views

What interesting homotopy invariants can I write down using the universal property of homotopy types?

I've recently been led to believe some version of the following statement: Weak homotopy types, or equivalently $\infty$-groupoids (let me not commit myself to a particular model of these), are ...
9
votes
4answers
627 views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
5
votes
1answer
202 views

What's the link between topological spaces as locales and topological spaces as infinity-groupoids?

I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...
3
votes
0answers
154 views

About a Double-pseudo-category generalization of the module bicategory construction

To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is ...
8
votes
1answer
263 views

Loops and suspensions of higher categories

Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ ...
10
votes
1answer
626 views

Lower Algebra: Modules over the monoidal category of abelian groups

Proposition 6.3.2.18 of Higher Algebra identifies $Mod_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable ...
9
votes
1answer
321 views

The state of the art in the rectification of homotopy-coherent structures

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a ...
3
votes
0answers
204 views

For an $(\infty,1)$-topos, is the object functor from groupoid objects to the topos a fibration, cofibration?

This question is kind of suggested by the question Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects? Question: If $\mathcal C$ in an ...
5
votes
1answer
316 views

Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?

Let $C$ be an $(\infty,1)$-topos. The $(\infty,1)$-category of group objects in $C$ is a full sub-$(\infty,1)$-category of groupoid objects in $C$: $${\mathsf{Grp}}(C) \hookrightarrow ...
8
votes
1answer
256 views

Is there a Wall finiteness obstruction in other settings?

Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW ...
21
votes
0answers
659 views

categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result? Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...
3
votes
0answers
265 views

In the topos-theoretic interpretation of Physics by Isham & Doering what role does intuitionistic logic play? [closed]

I've asked this question on Physics.SE but was advised to ask it here. Isham & Doering have written a series of papers exploring how to ground physics in topoi. Now the internal logic of topoi is ...
6
votes
0answers
165 views

Compact objects in undercategories and filtered colimits

Let $\mathcal{C}$ be a compactly generated presentable $(\infty, 1)$-category. Consider the functor $$ \Phi: \mathcal{C} \to \mathrm{Cat}_\infty, \quad x \mapsto (\mathcal{C}_{x/})^\omega,$$ that ...
2
votes
0answers
265 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
272 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
180 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 ...
17
votes
2answers
504 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
283 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
226 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 ...
35
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 ...
19
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
376 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
172 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
328 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
401 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
435 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
384 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
225 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
312 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
79 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
379 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
743 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
451 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 ...
9
votes
1answer
619 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
261 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 ...