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Thanks for pointing out any reference.

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Is there a specific setting in which you want to study homotopy (co)invariants? For example, do you have in mind a question about topological spaces, or complexes of vector spaces? Are you looking for an abstract homotopy-theoretic treatment? –  S. Carnahan Nov 1 '10 at 16:39
    
In Costello's work "The Gromov-Witten potential associated to a TCFT", it says the pullback operation is not well-defined in simplicial chain, but it is defined for homotopy coinvariant, that's what I am studying. –  HYYY Nov 14 '10 at 9:52

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If you like complexes of vector spaces over the field with two elements, Lecture 2 in this set of notes by Jacob Lurie has a nice treatment of homotopy coinvariants in that context. Some technical complexity that arises in the topological category is avoided in the linear algebraic treatment.

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Analogues of coinvariants for comodule algebras over Hopf algebras in model categorical setup is in a recent paper of Kathryn Hess on the arXiv. –  Zoran Skoda Nov 1 '10 at 9:38

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