# Tagged Questions

Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas ...

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### A definition of the homotopy colimit of a coherent diagram

Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...
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### Homotopy classification of selfmaps of product of spheres?

Self-maps of n-torus $T^n=S^1\times ...\times S^1$ are classified by the induced homorphism of fundamental group $\pi_1 T^n=Z^n$. Is a similar result true form self-maps of $S^k\times ...\times S^k$ (...
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### Principal bundles that can't be detected by spheres

The question I'm trying to answer is the following: Let $P \to X$ be a principal $G$-bundle (over a connected CW complex) satisfying that all pullbacks to spheres (of arbitrary dimension) are ...
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### Algebraic K theory, Karoubi completion and splitting

Suppose $\mathcal{C}\subset \mathcal{C}'$ is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, $\mathbb{C}$), such that $\mathcal{C}$ is faithfully embedded in ...
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### When does algebraic K theory behave like a cohomology theory

Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (...
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### Automorphisms of Eilenberg-Mac Lane spaces and semidirect products (and the odd line)

If $A$ is an abelian group, we have $Aut\left(K\left(A,n\right)\right)=Aut(A) \ltimes K\left(A,n\right),$ where the left hand side is the space of self-homotopy equivalences. Is there an easy way to ...
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### Symmetric sub-simplicial sets of the nerve of $\mathbb{N}$

The nerve of $\mathbb{N}$ is the simplicial set $N\mathbb{N}$ with simplices: tuples $(k_1,\ldots, k_r)$ with each $k_i\in \mathbb{N}$ degeneracies: inserting $0$ faces: adding consecutive entries ...
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### In CGWH, is every cofibration an inclusion with closed image?

As the title suggests, in CGWH, is every cofibration an inclusion with closed image?
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### What are some examples of total derived functors that can't be computed from a functorial replacement?

(Or more generally, what are some examples of Kan extensions which are not pointwise?) By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...
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### Generalization for Leray Hirsch theorem for Principal $G$-bundle [closed]

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
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### Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this. I wonder if there is any source telling about a ...
Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories? That is, for each spectrum $E$, do we have a lift in the ...
Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice \$\...