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4
votes
1answer
126 views

On closed model categories: standard arguments and fibrantly cogenerated categories

Some not very clever questions on closed model categories. For a (left or right) Quillen functor $F:C\to D$ what arguments does one usually use for proving that $Ho F$ is fully faithful when ...
4
votes
1answer
131 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 ...
4
votes
0answers
327 views

Interaction petit topos - gros topos

I know that there are already several discussions on this topic (and they already helped me a lot, so far), but I couldn't find one completely solving my problem. Question 1. Fix a scheme $X$. I know ...
3
votes
1answer
278 views

Maps between classifying spaces

Let $G$ be a discrete group and let $BG \simeq K(G,1)$ be its classifying space. Let $H$ be a topological group with classifying space $BH$. In case $H$ is also discrete, it was pointed out in the ...
3
votes
0answers
156 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 ...
3
votes
1answer
230 views

$T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?

I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through ...
9
votes
0answers
220 views

The spectral sequence of a tower of principal fibrations

Assume we have a tower of fibrations (of simplicial sets, let's say): $$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$ Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...
2
votes
3answers
240 views

How to detect if a simplicial set is the nerve of a groupoid?

I have the following question. Suppose I have a simplicial set. Is there a way to detect if it actually is isomorphic to a nerve of a groupoid? I've seen the fact that if you have a nerve ...
4
votes
0answers
158 views

Homotopy unites of a differential graded algebra

I apologize in advance if the question is too basic. Let $A$ be a differential graded algebra and $H^{0}A$ the 0-cohomology of $A$ (which is an ordinary ring) and $A^0$ is the 0-level of $ A$ . An ...
0
votes
0answers
54 views

can we say fixed point existance of a set valued map over a compact set is homotopy invariant?

Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair ...
9
votes
1answer
498 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 ...
4
votes
1answer
175 views

How does one calculate homotopy classes for group coset spaces?

Inspired by Witten's Wess-Zumino term arguments, I'm curious to know how one calculates homotopy classes more generally for coset spaces. In the above example the coset is $G/H=(SU(3)_L\times ...
6
votes
2answers
297 views

Must the union of these two aspherical spaces be aspherical?

Let $X$ be a reasonably nice topological space (say, a connected CW complex), and let $Y$ and $Z$ be reasonably nice connected subspaces of $X$ such that $X = Y \cup Z$. Suppose that $Y$, $Z$, and ...
3
votes
0answers
168 views

What does it mean for the $\Pi_\infty$ groupoid to be fully fibrant?

The original question I had was: If I have a sequence of simplicial spaces $$A\to B\to C$$ which is degree-wise a homotopy fibration, under which conditions is the geometric ...
5
votes
1answer
366 views

Finite homotopy limits commute with sequential homotopy colimits

I would like to know for what kind of model category finite homotopy limits commute with sequential homotopy colimits. Would cofibrantly generated and finitely locally presentable be enough? It seems ...
7
votes
1answer
399 views

Homotopy groups of an infinite wedge of 2-spheres

I know Hilton's result about a finite wedge of spheres, and I know that certain homotopy groups (such as the third homotopy group) can be directly calculated for an infinite wedge too. My question is ...
0
votes
2answers
287 views

even dimensional manifold homotopic to a symplectic or complex manifold

I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a ...
4
votes
1answer
484 views

Topological Hochschild cohomology?

Let $A$ be a $E_\infty$-ring spectrum. By EKMM, it may be treated as a commutative algebra in the appropriate category. In particular, one may define topological Hochschild homology as ...
5
votes
1answer
576 views

A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a regular CW complex of constant local dimension $n$. $X_{n}$ is of finite type, ...
4
votes
2answers
428 views

On triangulated categories of pro-objects

Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories? I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...
1
vote
1answer
153 views

isotopy classes of embeddings of the torus

Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$? For each free homotopy classes $\gamma$ of mappings of the circle ...
2
votes
0answers
202 views

A topos-theoretic thickening of the nerve functor

Let $\Delta$ be the standard cosimplicial space sending $[n]\mapsto \Delta^n\subset \mathbb R^{n+1}$. Then the correspondence $\delta\colon [n]\mapsto {\rm Sh}(\Delta^n)$ defines a cosimplicial ...
7
votes
0answers
233 views

The spheres operad

I have a rather naive question. Consider the space of all maps $$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$ for all possible natural numbers $n, k, j_1, \cdots , j_k$. This ...
6
votes
3answers
308 views

Cobordism and finite sheeted covers of manifolds

Let $M$ be an oriented manifold, not necessarily compact. Let $M'$ be a (finite) $k$-sheeted cover and let $\pi:M'\longrightarrow M$ be the covering map. Question 1 : Is it true that $M'$ is ...
8
votes
1answer
252 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 $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ ...
4
votes
2answers
295 views

P.J. Hilton notes requested

Does anybody here have the mimeographed notes Homotopy theory and duality, by P.J. Hilton, Cornell University, 1959 ? I guess that those notes were never published online. I believe that some ...
8
votes
1answer
281 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 ...
0
votes
0answers
136 views

The nerve-realization of $[n]\mapsto\Pi_1(\Delta^n)$

Consider the diagram Where the functor $G$ sends a topological space to the category having as objects its points, and arrows homotopy classes of paths, $\varrho$ "realizes" geometrically an ...
7
votes
1answer
233 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 ...
4
votes
3answers
519 views

Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below : This post has been divided into two parts, the second part is here. Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
6
votes
2answers
469 views

Homotopy problem for infinite dimensional topological space II

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying : $X_{n}$ have topological dimension $n$. $X_{n+1}$ is ...
0
votes
2answers
279 views

Homotopy problem for infinite dimensional topological space

Let $X$ be an infinite dimensional topological space such that : $ \forall n \in \mathbb{N}$, $ \exists X_{n} \subset X$, $n$-dimensional subspaces verifying : $\forall r<n$, the homotopy ...
8
votes
2answers
404 views

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

A $L_\infty$-algebra can be defined in many different ways. One common way, that gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
8
votes
2answers
536 views

Homology of localisations of spectra

Let $H^*$ and $K^*$ be two cohomology theories, and $X$ a reasonable spectrum. Here, I'm thinking that $H^*$ is singular cohomology (and for my purposes, rational cohomology will suffice), and $K$ is ...
5
votes
0answers
182 views

Geometric realization of simplicial spaces and finite limits

Let $X_{\bullet}$ be a simplicial space and denote the (non-fat!) geometric realization by $\lvert X_{\bullet} \rvert$. Does this geometric realization of simplicial spaces preserve finite ...
16
votes
2answers
507 views

Conceptual explanation for the relationship between Clifford algebras and KO

Recall the following table of Clifford algebras: $$\begin{array}{ccc} n & Cl_n & M_n/i^{*}M_{n+1}\\ 1 & \mathbb{C} & \mathbb{Z}/2\mathbb{Z} \\ 2 & \mathbb{H} & ...
10
votes
0answers
248 views

When is the one-point compactification well-pointed?

This is a follow up to my previous question. Question: Is there a reasonably natural set of conditions which guarantee that the one-point compactification $X^+$ of a locally compact Hausdorff ...
17
votes
1answer
843 views

The cell structure of Thom spectra

I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a ...
5
votes
1answer
193 views

Pontrjagin ring structure on homology of Eilenberg-Mac Lane spaces

Is there any good reference for the Pontrjagin ring structure on $$ H_\ast(K(\mathbb{Z}/2,k);\mathbb{Z}/2)\cong H_\ast(\Omega K(\mathbb{Z}/2,k+1);\mathbb{Z}/2)? $$ I am familiar with Serre's theorem ...
3
votes
0answers
179 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 ...
6
votes
2answers
228 views

Components of a loop space, semidirect products, and multiplicativity

Let $(X, x_0)$ be a based topological space, and $\Omega X$ its based loop space. The group of path components of $\Omega X$ is $\pi_0(\Omega X) = \pi_1(X, x_0)$. For brevity, let's call this group ...
4
votes
0answers
170 views

Adding morphisms to a category without changing homotopy type

I have a really tame category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is ...
2
votes
0answers
110 views

Twisted product structure on product of Eilenberg-MacLane spaces

The product of Eilenberg-MacLane spaces $K(\mathbb{Z}/2,1) \times K(\mathbb{Z}/2,2)$ is an $H$-space with respect to a "twisted product structure", where you add the product of the first entries to ...
4
votes
3answers
677 views

When are maps between topological spaces homotopic?

I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say). So far I had the following ...
5
votes
0answers
133 views

Refined cofinality theorem for homotopy limits of spaces

If $C$ is a simplicial model category, $F\colon I\longrightarrow J$ a functor between small categories, and $X\colon J\longrightarrow C$ a diagram, the canonical map $holim_JX\longrightarrow ...
4
votes
0answers
213 views

Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...
13
votes
3answers
556 views

What are finite homotopy types?

Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’: The homotopy type of a simplicial set that has only finitely many ...
32
votes
9answers
2k views

Homotopy as a general organizing principle

One of the realizations that led to the development of Homotopy Type Theory (HoTT) is that the ideas of homotopy theory have very broad applicability in mathematics. Indeed, Quillen model categories ...
3
votes
1answer
183 views

Bousfield localization before and after taking homotopy

The ncatlab says: Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...
19
votes
2answers
1k views

How does it End?

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context. Let $\mathcal{C}$ be a category. ...