Nonabelian Algebraic Models for Classical Homotopy Types

The goal of this dissertation to establish a new category of algebraic models for homotopy types, by replacing the simplicial category Ord with the symmetric-simplicial category Fin in the equivalences (1) between the homotopy category of spaces and that of simplicial sets; (2) between the homotopy category of reduced simplicial sets and that of simplicial groups; and (3) between the category of simplicial groups and that of the hypercrossed complexes of Carrasco-Cegarra. The symmetric version of (1) is a consequence of an old result of Dwyer-Hopkins-Kan, whereas the symmetric versions of (2) and (3) appear to be new. The motivation for doing this appears in our study of certain symmetric analogs of hypercrossed complexes, whose data are better organized than in ordinary hypercrossed complexes. Meanwhile we provide algorithms for generating the axioms of hypercrossed complexes and higher semidirect products in general. We also generalize the nonabelian Dold-Kan decomposition Carrasco-Cegarra, giving a class of decompositions which we believe is exhaustive (although we do not prove this). For symmetric hypercrossed complexes we also give a complete closed-form enumeration of the axioms corresponding to associativity of the corresponding symmetric-simplicial group (this seems to be impossible for general hypercrossed complexes). In order to describe the model category structure on symmetric-simplicial groups, we construct an explicit deformation retraction of the free simplicial group on a reduced simplex onto the free simplicial group on a reduced horn, thus giving a concrete geometric interpretation to the well-known fact that every simplicial group possesses the Kan extension property. The retractions are given by highly nonabelian formulas which are essentially the same as formulas given by Carrasco-Cegarra for the projections in their nonabelian Dold-Kan decomposition. To prove the Quillen equivalence between reduced symmetric-simplicial sets and symmetric-simplicial groups, we are obliged to extend the machinery of principle bundles and classifying spaces from the simplicial to the symmetric-simplicial setting. It is remarkable that a single set of formulas suffices for either setting. On the other hand, the phenomena arising in the proof that the symmetric-simplicial loop-group functor preserves cofibrations have no straightforward analogs in the simplicial case.