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I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain complexes such that the identities between cup and cap products hold.

I was thinking about the comonoid objects in the closed monoidal categories, where you can create caps and cups, but to have standart $identities$ between caps and cups as is in the situation of the singular chain complexes it is not enough to have triangle and cocomutativity coherence conditions for the comonoid object $C$. So I am interested to know better framework.

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  • $\begingroup$ Well, what I have in mind is the diagonal approximation for the singular chain complex in as a particular case of comonoid object in the homotopy category of chain complexes. Then you have several nice identities relating caps , cups and evaluations in this particular case. That was the motivation for the question. $\endgroup$ Nov 12, 2014 at 23:58
  • $\begingroup$ You're moving in a less general framework so the should be no more conditions. $\endgroup$ Nov 13, 2014 at 0:03
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    $\begingroup$ Let me put it this way: some coherence conditions which would guarantee analogs of identities between caps and cups in the case of diagonal approximations for the singular chain complex. $\endgroup$ Nov 13, 2014 at 0:05
  • $\begingroup$ @DmitriScheglov that is extra structure, and not extra properties. Maybe you should reformulate your whole question. BTW, Tsygan-Tamarkin's notion of 'calculus' may be interesting for you. $\endgroup$ Nov 13, 2014 at 8:24
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    $\begingroup$ ncatlab.org/nlab/show/noncommutative%20differential%20calculus $\endgroup$ Nov 13, 2014 at 14:23

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