Questions tagged [brown-representability]

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Is cohomology with local coefficients a representable functor?

It is well known that the functor of cohomology is representable. More precisely, given $n\ge1$ and abelian group $G$, we have $H^n(X;G)\simeq[X,K(G,n)]$. (Here we probably need some ``nice'' ...
Andrey Ryabichev's user avatar
3 votes
0 answers

Brown Representability for Stable Homotopy Categories of Symmetric Spectra

Proposition 5.5 in $\mathbf{A}^1$-homotopy theory establishes the Brown's representability for the stable homotopy category $\mathcal{SH}_T(S)$, over a Noetherian scheme $S$, for a space $T$ of finite ...
user24453's user avatar
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2 votes
2 answers

Is there always a universal bundle over a classifying space?

Consider some kind of bundles, for instance vector bundles or fibre bundles with a certain structure group, such that there are bundle morphisms and pull-backs. Let then $F(X)$ denote the ...
Jan Steinebrunner's user avatar
4 votes
2 answers

A CW is of countable type, iff all its homotopy groups are countable? (References?)

When constructing a classifying space $BPL$ for piecewise linear microbundles, one would like it to be a polyhedron, i.e. a locally finite simplicial complex. Milnor solved this by showing that the ...
Jan Steinebrunner's user avatar
59 votes
2 answers

Are spectra really the same as cohomology theories?

Let $E \to F$ be a morphism of cohomology theories defined on finite CW complexes. Then by Brown representability, $E, F$ are represented by spectra, and the map $E \to F$ comes from a map of spectra. ...
Akhil Mathew's user avatar
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47 votes
1 answer

Brown representability for non-connected spaces

In many places (on MO, elsewhere on the Internet, and perhaps even in some textbooks) one finds a statement of the classical Brown representability theorem that looks something like this: If $F$ is ...
Mike Shulman's user avatar
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