7
votes
0answers
204 views

I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category?

I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need ...
-1
votes
0answers
154 views

Algebraic topology vs. category theory [migrated]

I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...
1
vote
1answer
176 views

Pushout of categories along embeddings gives homotopy pushout?

Consider the pushout of a diagram $A\leftarrow B\rightarrow C$ of categories and assume that at least one of the arrows is an embedding, i.e. injective on objects and arrows. When applying the nerve ...
8
votes
1answer
299 views

Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid

Let $C$ be a category with an object $X$ such that there are no non-trivial endomorphisms $X\rightarrow X$. Consider a simplex $\sigma$ of the nerve $NC$ of $C$. It is just a string of composable ...
1
vote
1answer
308 views

A question about the proof of Quillen's Theorem A

(I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.) Theorem (Quillen) ...
12
votes
4answers
603 views

Functors and coverings

A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical ...
6
votes
1answer
266 views

Can the monoidal structure on manifolds be strictified?

I'm asking this question purely out of curiosity. Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...
2
votes
0answers
204 views

Why does this setting imply that a category is Grothendieck?

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf. Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...
21
votes
5answers
1k views

(Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting. The problem is to produce an example of the following ...
1
vote
1answer
135 views

comparison between two monadic definitions for an operad

According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$ According to Leinster, an operad is ...
4
votes
0answers
182 views

Limits and colimits of A_{\infty} categories

I have a question related to the discussion (Coequalizer in category of dg-algebras). How do you prove that the category of (small) dg-categories and the category of (small) A_{\infty} categories are ...
7
votes
1answer
193 views

Correspondence between operads and monads requires tensor distribute over coproduct?

In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...
1
vote
0answers
210 views

Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...
2
votes
1answer
279 views

Homotopy groups of filtered homotopy limits

Let $X$ be the homotopy limit of a filtered system of simplicial sets $X_i$. When are the morphisms $\pi_j(X)\to \varprojlim \pi_j(X_i)$ surjective for all $j\ge 0$? This seems to be no problem when ...
3
votes
0answers
115 views

What are the Čech-local equivalences of (simplicial pre)sheaves?

Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield ...
1
vote
1answer
276 views

A $2$-torsion version of the motivic stable homotopy category?

For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category ...
5
votes
0answers
146 views

Weak equivalences of left Bousfield localizations

Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1. If necessary, the model structures can be assumed to be simplicial, ...
8
votes
1answer
274 views

Which statements and arguments of Hovey's “Model categories” fail without functorial factorizations of morphisms?

I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations ...
3
votes
1answer
155 views

Factorization of morphisms in a diagram category

Let us suppose that $I$ is a small category and $\mathcal{E}$ a combinatorial model category. Then there exists two Quillen equivalent combinatorial model category structures on the diagram category ...
3
votes
2answers
225 views

Analogues of 'cone' distinguished triangles for pointed model categories?

For an additive $A$ and any morphism $f:X\to Y$ in $C(A)$ one has the following distinguished triangle in the homotopy category $K(A)$: $X\to Y\to Cone(f)\to X[1]$. What is the closest analogue of ...
5
votes
4answers
423 views

Configuration topos?

Let ${\bf Fin}$ denote the category of finite sets. If $X$ is a topological space, then for any natural number $k\in{\mathbb N}$, the slice category ${\bf Fin}/X$ contains the configuration space ...
4
votes
1answer
128 views

Yoneda embeddings of stable model categories; composition with Bousfield localizations

For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question ...
3
votes
0answers
139 views

Homotopy theory of acyclic categories

Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the ...
9
votes
1answer
484 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 ...
2
votes
1answer
166 views

Left homotopy in the quillen model structure

Let $ \mathscr{T} $ be the category of [insert technical conditions here] topological spaces. Equip $ \mathscr{T}$ with the quillen model structure. The category of based spaces $ \mathscr{T}_*$ ...
8
votes
1answer
217 views

Does a bifunctor that's monoidal in each argument take pairs of monoids to a commutative monoid?

Let $\mathcal{C}, \mathcal{D}, \mathcal{E}$ be (symmetric?) monoidal categories, and $H : \mathcal{C} \times \mathcal{D} \to \mathcal{E}$ be a functor that is monoidal in both arguments, ie. $H(C,-)$ ...
0
votes
0answers
188 views

is this a simplicial model category?

A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...
1
vote
1answer
216 views

Standard way to prove that groupoids are homotopy 1-types

It is very well know that groupoids, considered as spaces via the nerve construction, are homotopy 1-types, i.e. aspherical. Here is a sketch of proof: Consider the canonical functor $f:C\rightarrow ...
7
votes
1answer
226 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 ...
6
votes
2answers
505 views

Aspherical operads

Let $O$ be an operad in $\mathtt{SETS}$. Assume that $O(0)$ is empty and $O(1)$ only consists of the identity. Assume for simplicity that $O$ is monochromatic, i.e. we have no labels on the ...
4
votes
1answer
194 views

U(1) vs. BZ and representations of 2-groups

$U(1)$ seems to lead a dual life. On one hand it is the group we know and love, and on the other, it is the classifying space of the integers. Thinking about $n$-groups says that we should also think ...
3
votes
0answers
175 views

Are there CW structures on homotopy limits of CW maps?

Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its homotopy colimit is precisely the mapping cylinder $$C_H = \frac{X \sqcup (Y ...
1
vote
1answer
138 views

A $G$-space as a coend

Let $\bf Top$ a convenient category of topological spaces, $G$ a group in $\bf Top$, ${}^G\bf Top$ the category of (left) $G$-spaces, and $Sgrp(G)$ the poset of subgroups of $G$. Define two functors ...
7
votes
0answers
195 views

Two model categories I would like to know if they are Quillen equivalent or not

It is the motivation of the question Examples of non Quillen-equivalent model categories having equivalent homotopy categories. I did not give at first the motivation because i don't think that people ...
6
votes
2answers
343 views

Examples of non Quillen-equivalent model categories having equivalent homotopy categories

I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories. The motivation is of course that I have two model categories and all ...
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 ...
1
vote
1answer
157 views

Free groupoid and homotopy equivalence

Let $C$ be a (small) category. One can form the free groupoid $GC$ of $C$ which is the left adjoint construction to the inclusion functor $\mathrm{Groupoid}\rightarrow\mathrm{Category}$. Is then $C$ ...
3
votes
1answer
151 views

Directed colimits of maps in a combinatorial model category

I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated ...
7
votes
1answer
434 views

Grothendieck fibrations and classifying spaces

Suppose that $$F:\mathcal{D} \to \mathcal{C}$$ is a Grothendieck fibration of small categories, whose fibers are groupoids. Is there anything sensible which can be said about the induced morphism ...
0
votes
0answers
187 views

Homology of the dg-nerve vs Hochschild homology of the dg-category

Lurie in Higher Algebra, section 1.3 associates a quasi-category to a dg-category A via the so called dg-nerve construction, extending the classical nerve. I have a feeling the homology of the ...
3
votes
1answer
218 views

Resolutions chain homotopic to projective ones

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module ...
3
votes
0answers
181 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
272 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 ...
1
vote
1answer
232 views

Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...
4
votes
2answers
739 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 ...
7
votes
6answers
1k views

Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here. Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$. Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...
1
vote
1answer
123 views

Brown representability for the standard model category of simplicial sets

Let $Sset$ denote the category of simplicial sets with its Quillen model structure, when is a functor $F: (ho Sset)^{op} \to Ab$ representable? With $Ab$ category of Abelian groups. There is probably ...
2
votes
1answer
223 views

Path components of a monoidal category acting on homology

Let $S$ be a (small) symmetric monoidal category and $X$ a (small) category on which $S$ acts. $\pi_0(S) = \pi_0(BS)$ is naturally an abelian monoid, with $[A] + [B] := [A+B]$, where $[A]$ denotes ...
2
votes
2answers
211 views

Finitely cocomplete categories of compact Hausdorff spaces

Edit: Zhen Lin incisively observes in a comment below that the category of compact Hausdorff spaces is monadic over the category of sets, hence is cocomplete. That answers the first part of question 1 ...
7
votes
2answers
237 views

Can one compare monads arising from homotopy equivalent adjunctions?

Suppose we have a Quillen adjunction $L\colon {\mathcal C} \leftrightarrow {\mathcal D}: R$. For convenience let us assume that all objects of $\mathcal C$ are cofibrant and all objects of $\mathcal ...