3
votes
1answer
236 views

Classifying space for fibrations with Eilenberg-MacLane space as fibers

The following result seems to be frequently quoted: Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). ...
5
votes
0answers
89 views

Explicit generators for homotopy groups of Lie groups

I would like to know explicit formulas for generators of the infinite cyclic summands in the homotopy groups of Lie groups, in the form of continuous (or smooth if possible) maps $S^n\to G$. It is ...
10
votes
2answers
395 views

“Economic” CW-structure for Eilenberg-MacLane spaces?

The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even ...
3
votes
0answers
122 views

Flat Connections on the Cotangent Complex

I'm trying to find a reference which defines and discusses some properties of connections and flat connections on the cotangent complex in a homotopical setting. That is to say, a connection or flat ...
4
votes
1answer
134 views

In a Simplicial group the Degenerate elements don't intersect the kernel of the face maps.

Let $G_{\bullet}$ be a simplical group. I denote by $D_n \subset G_n$ the subgroup generated by all the degeneracy maps $s_i:G_{n-1} \to G_n$. I also denote $$M_n = \bigcap_{i>0} \mathrm{Ker} d_i ...
2
votes
1answer
275 views

The topology of the classifying space of U(n)

In $\textit{Atiyah and Bott: The Yang-Mills equations over riemann surfaces}$, there is a sentence about the topology of BU(n) (i.e. the classifying space of U(n)): Here $K(\mathbb{Z},n)$ means the ...
4
votes
2answers
319 views

Aspheric functors and Grothendieck fibrations

Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, ...
3
votes
2answers
347 views

Zigzags and contractibility of categories

Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion: If there is ...
5
votes
0answers
217 views

Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let $q: D^k \to D^r$ be a map and $r \leq k$. Let $$W = S^k \sqcup D^r/\sim$$ where ...
1
vote
0answers
101 views

Retracts of 2-categories

Let $\mathbf{C}$ be a strict $2$-category and let $M = \{(x_\alpha,y_\alpha)\}$ be a collection of object-pairs so that each hom-category $\mathbf{C}(x_\alpha,y_\alpha)$ has an initial element ...
6
votes
1answer
229 views

Group actions in a homotopy category

Let $M$ be a model category and $G$ a finite group, and equip the category $M^G$ of $G$-objects in $M$ with, say, a projective model structure. Then there is a canonical functor $$\mathrm{Ho}(M^G) ...
8
votes
1answer
226 views

Shriek push-forward for parameterized spectra

In May and Sigurdsson's Parameterized Homotopy Theory, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow ...
4
votes
0answers
170 views

“Naïve”cobordism?

The naive view of (say, complex) cobordism of a space $X$ is that it should be the group generated by continuous families of closed manifolds parametrized by $X$. Acting even more naively, one may try ...
3
votes
0answers
115 views

Infinity category of functors from a relative category to a model category

Let $M$ be a model category (maybe cofibrantly generated/combinatorial). Let $(C,W)$ be a relative category. I write $M^{(C,W)}$ for the full subcategory of $M^C$ on relative functors. This is a ...
8
votes
3answers
935 views

Is there a homology theory that gives a *necessary and sufficient* condition for homotopy equivalence?

Is there a (non-Abelian) homology theory that realizes the following: Let $X,Y$ be manifolds with complexes $C(X),C(Y)$. Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ ...
1
vote
1answer
206 views

Thomason’s homotopy colimit theorem for pseudo-functor

Thomason’s homotopy colimit theorem R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Camb. Phil. Soc. (1) 85 (1979)91-109 says that for a functor $F:I^{op}\to ...
2
votes
1answer
223 views

Massey product in Dual Steenrod Algebra

Let $\tau_0$ be the element of dual Steenrod algebra $A_p^{*}$ at a prime $p$ which is dual to Bockstein $\beta \in A_p$. It is well known $\tau_0^2 =0$. Is it true/known that the elemnet $\xi_1$ ...
3
votes
1answer
270 views

What is the “higher version” of chain homotopy in singular homology?

In basic algebraic topology, we know the following well-known chain homotopy theorem: Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...
2
votes
0answers
154 views

Quillen functors and stable model categories

Are there any books or papers where I can find some general statements and methods for working with Quillen functors that are not equivalences (and not localizations)? In particular, I would like to ...
4
votes
1answer
128 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 ...
3
votes
0answers
160 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 ...
4
votes
1answer
238 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 ...
8
votes
1answer
289 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 ...
5
votes
1answer
198 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
1answer
187 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 ...
4
votes
4answers
483 views

What is the homotopy fiber of a fold map?

If $X$ and $Y$ are based spaces, let $p_X: X\vee Y\to X$ be the fold map, or projection, onto $X$. What is the homotopy fiber $F$ of $p_X$? I think I have an argument that $F$ is the half-smash ...
6
votes
5answers
586 views

What is a good basic reference on model categories?

I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ...
4
votes
1answer
372 views

Borel constructions, equivariant cohomology, and homotopy quotients of monoid actions.

Let $M$ be a (discrete) monoid acting on a space $X.$ We may take the quotient of $X,$ by this action, $X/M$ that is the coequalizer of the action map $M \times X \to X$ against the projection map. ...
4
votes
1answer
356 views

“monotone” homotopy?

This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations. Let $X$ be a (bounded) metric space, $Y$ be a topological space and ...
6
votes
0answers
188 views

What is the state of the art in 4-manfold 2-types?

In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...
4
votes
3answers
437 views

Monoidal model category structure on a functor category.

Let $A$ be a small simplicial category. The category $Fun(A,s\mathrm{Set})$ of simplicial functors from $A$ to simplicial sets can be given the projective model structure in which fibration and weak ...
7
votes
2answers
437 views

topological monoid from symmetric monoidal category

What is the standard reference for the fact that the classifying space of a strict monoidal category is a topological monoid with respect to the operation induced by the tensor product? EDIT: The ...
11
votes
2answers
651 views

What is the homotopy type of the space of the homeomorphisms of the n-ball such that the homeomorphism restricted to the boundary is isotopic to the identity?

Consider the set of homeomorphisms of the topological n-ball to itself with the compact open topology. Sitting inside this space of homeomorphisms are particular subspaces. The first subspace is those ...
3
votes
2answers
254 views

Second homotopy groups of 3-complexes and Fenn's spiders.

Let $X$ be a finite CW complex then with one zero cell. Then (up to homotopy) the two skeleton of X is the same as a group presentation, via the Cayley complex construction. For a while I had been ...
3
votes
3answers
334 views

Second homotopy group of Cayley complex

Is there a good reference for information about the second homotopy group of the Cayley complex or Presentation complex of a finitely presented group, especially a hyperbolic group? I'm looking for ...
8
votes
4answers
2k views

Lurie's “Virtual fundamental classes” and “Geometric derived stacks”

In his thesis, Jacob Lurie mentioned two work in preparation (by him), namely "Virtual fundamental classes and the motivic sphere" and "Geometric derived stacks". Now that much is written in the DAG ...
3
votes
1answer
197 views

Explicit contraction for the universal simplicial bundle WG

For a simplicial group $G$, there is a universal bundle $WG \to \overline{W}G$ in the category of simplicial sets, detailed in for example May's book (djvu). Now $WG$ has a simple enough description ...
2
votes
2answers
304 views

Connected covering spaces of a homotopy colimit

Let $\mathcal{D}: C\to Top$ be a diagram of spaces (spaces are "nice", and $C$ is small). Let $X$ denote the homotopy colimit of $\mathcal{D}$ (which is connected) and $\pi(C)$ be the free groupoid on ...
10
votes
2answers
510 views

Difficulties with the mod 2 Moore Spectrum

I have been informed that there is a reference out there which specifically details what goes wrong with the mod 2 Moore spectrum, i.e. why it is not $A_\infty$ or something? I do not know the ...
17
votes
2answers
1k views

Surveys of Goodwillie Calculus

Is there a good general introduction to Goodwillie calculus out there, like a paper or publication that gives a general overview of the calculus as well as how it is useful and why we are interested ...
15
votes
2answers
911 views

Proof of Bott Periodicity in twisted K-theory

I have a question about the Proof of Bott Periodicity in twisted K-theory by Atiyah and Segal in their paper Twisted K-theory. Following their notation, to prove Bott periodicity in this context it ...
4
votes
1answer
272 views

Homotopy dimension of a mapping

The homotopy dimension $h \dim X$ of a space $X$ is the minimal dimension of a CW-complex $Z$ homotopy equivalent to $X$. I am interested in the generalisation to maps $f\colon X\to Y$. Here's what I ...
6
votes
1answer
309 views

How equivalent are the theories of reduced and groupal $\infty$-groupoids?

I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this ...
6
votes
3answers
290 views

Order of the identity map of a Moore space.

Write $M_n = S^n \cup_2 D^{n+1}$. I know, as a matter of folklore, that the identity map $\mathrm{id}_M$, considered as an element of the group $[M_n, M_n]$, has order $4$ (for $n > 3$, let's ...
1
vote
1answer
194 views

James Construction for Disconnected Spaces

When I work out the James construction for a discrete pointed space, it appears that the induced map $\pi_0 (J(X)) \to \pi_0( \Omega\Sigma X)$ is the inclusion of the free monoid on $\pi_0(X)$ into ...
5
votes
3answers
541 views

A reference for Calculus of Functors for Model Categories

I am wondering where I might look to see what has been done in terms of Calculus of Functors for more general weak equivalences and Model Categories. I am at least aware of some of the extended ...
2
votes
0answers
424 views

How to prove that a map is a Serre fibration?

I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that ...
11
votes
1answer
428 views

The space of compact subspaces of $R^\infty$ homotopy equivalent to a given finite complex.

Let $X$ be a finite (CW or simplicial - doesn't matter) complex and consider the space of all compact subspaces of $R^\infty$ which are homotopy equivalent to $X$, topologized say as a subspace of the ...
14
votes
1answer
781 views

Comodule exercises desired

This Question is inspired by a Quote of Moore's "There are two ‘evil’ influences at work here: 1. we are toilet trained with algebras not coalgebras 2. some of us are addicted to manifolds and so ...