Injective models of \(G\)-disconnected simplicial sets. (English) Zbl 0886.55012

The author extends the rational homotopy theory on \(G\)-disconnected simplicial sets being not necessary of finite \(G\)-type. Instead of the orbit category \(O(G)\), in this paper the author works over the category \(O(G,X)\) with one object for each component of each fixed point simplicial subset \(X^H\) of a \(G\)-simplicial set \(X\) for all subgroups \(H\) in \(G\). First he investigates the category \(kI\)-Mod of covariant functors on a small category \(I\) to the category of \(k\)-modules over a field \(k\) and presents some considerations about injective objects in the category \(kI\)-Mod. Then he extends these on the category \(I\)-\(DGA_k\) of functors from an \(EI\)-category \(I\) (all endomorphisms are isomorphisms) to the category \(DGA_k\). He presents the existence of an injective minimal model for a complete injective \(kI\)-algebra \(A\) for an \(EI\)-category \(I\). He shows that on de Rham algebra \(A^*_X\) of rational polynomial forms on a simplicial set \(X\) there is a natural complete linear topology. Finally, for a \(G\)-simplicial set \(X\) he associates the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\) \((\mathbb{Q}\) the field of rationals) and shows the existence of an injective minimal model for the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\). The main result describes the rational homotopy type of a nilpotent \(G\)-simplicial set \(X\) by means of injective minimal model of the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\).


55P62 Rational homotopy theory
55P91 Equivariant homotopy theory in algebraic topology
16W80 Topological and ordered rings and modules
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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