Questions tagged [brown-representability]

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. ...

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 ...

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'' ...

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 ...

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 ...

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 ...