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Results tagged with at.algebraic-topology
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user 11300
Homotopy theory, homological algebra, algebraic treatments of manifolds.
19
votes
Why chain homotopy when there is no topology in the background?
There is an inner-hom in $\mathbf{Chain}$, and the 1-chains are chain homotopies. The definition is
$$
\underline{\mathbf{Chain}}(C_\bullet,D_\bullet)_k = \Pi_n \textrm{Hom} (C_{n-k},D_n)
$$
so a 0-c …
- 569
4
votes
1
answer
248
views
Compatibility of classifying space with inner-hom?
Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category
$\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let …
- 569
4
votes
0
answers
206
views
Mapping into a geometric realization.
Suppose $S$ is a simplicial set, $X$ is a space, and we are given a map
\[
f: \text{Sing}\,X\to S.
\]
When is is possible to produce a map $X\to |S|$?
We can take the realization of $f$, to get $|f|: …
- 569
2
votes
1
answer
385
views
Embeddings of vector spaces
Let $V$ be an $n$-dimensional vector space. Is the space of embeddings
$$
\coprod_1^{k} V \to V
$$
path connected for large enough $n$? Clearly $n=1$ is not enough, but I feel like $n=2$ is enough f …
- 569