Thanks for pointing out any reference.

1$\begingroup$ 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 homotopytheoretic treatment? $\endgroup$ – S. Carnahan♦ Nov 1 '10 at 16:39

1$\begingroup$ In Costello's work "The GromovWitten potential associated to a TCFT", it says the pullback operation is not welldefined in simplicial chain, but it is defined for homotopy coinvariant, that's what I am studying. $\endgroup$ – HYYY Nov 14 '10 at 9:52
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.

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