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I don't remember if I've already seen this question even here or in MO or in my mind. This is partly related to questions arose about differences between homology and cohomology; I'm wondering if some kind of difference can appear before computing $H^n(-)$, $H_n(-)$, looking at the bare complex of (co)chains...

My question: Given a chain complex $\mathcal C=\{C_n,\partial_n\}$, consider the image complex $\hom(\mathcal C,X)=\{\hom(C_n,X),\hom(\partial_n,X)\}$; is it always a cochain complex? Can I always define some kind of ``equivalence'' between chain complexes and cochain complexes using a suitable contrav. functor? Can I conversely choose carefully $X$ in order not to have such an equivalence?

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    No matter what $X$ is and what sort of chain complexes you're looking at, applying the contravariant additive functor $\hom{({-},X)}$ will always result in a cochain complex (of abelian groups) since the relation $\partial_{n}\partial_{n+1} = 0$ is sent to the relation $\delta^{n+1} \delta^{n} = 0$, where I write $\delta^n = \hom{(\partial_n},X)$ for short. However, you might e.g. kill torsion: suppose for instance each $C_n$ is a torsion abelian group. If you take $X = \mathbb{Z}$ then $\hom{(C_n,X)} = 0$ for all $n$.2011-04-30
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    I don't see if it is true just because I'm taking $\hom$ or if it is true with any contrav. functor... I suppose there's anything peculiar in that functor. Good example that with torsion!2011-04-30
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    If $F$ is contravariant then $F(\partial_n \partial_{+1}) = F(\partial_{n+1})F(\partial_n)$. So if you want to send chain complexes to cochain complexes, you need contravariance and sending zero to zero. Since you're also interested in chain maps and the *additive* notion of (co)chain homotopy it seems best to restrict attention to *additive* functors.2011-04-30
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    On the other hand, there are not that many *contravariant* additive endofunctors of abelian groups by the contravariant version of the [Eilenberg-Watts theorem](http://ncatlab.org/nlab/show/Eilenberg-Watts+theorem) that says in particular that a contravariant additive endofunctor $F:\mathfrak{Ab} \to \mathfrak{Ab}$ has to be of the form $F({-}) = \hom{({-},F(\mathbb{Z}))}$, as soon as it transforms colimits into limits.2011-04-30
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    @t.b: If we knew $H_n$ was finite dimensional, and we were to apply $hom(-,X)$ twice, would we recover $H_n$?2014-06-12
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    @user99680: No, consider the example by t.b.2016-04-07

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