# 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'' ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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. ...

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### 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 ...