arXiv subjects

Eric R. Antokoletz

Publications and source records attributed to Eric R. Antokoletz.

3 recordsLinked to original sources

arXiv2010

An Alternative Presentation of the Symmetric-Simplicial Category

The category Fin of symmetric-simplicial operators is obtained by enlarging the category Ord of monotonic functions between the sets {0,1,...,n} to include all functions between the same sets. Marco Grandis has given a presentation of Fin using the standard generators Ord (cofaces and codegeneracies) as well as the adjacent transpositions which generate the permutations in Fin. The purpose of this note is to establish an alternative presentation of Fin in which the codegeneracies are replaced by special maps which we call quasi-codegeneracies. We also prove a unique factorization theorem for products of cofaces and quasi-codegeneracies analogous to the standard unique factorizations in Ord. This presentation has been used by the author to construct symmetric hypercrossed complexes (to be published elsewhere) which are algebraic models for homotopy types of spaces based on the hypercrossed complexes of Carrasco and Cegarra.

math.AT↗

arXiv2010

Nonabelian Dold-Kan Decompositions for Simplicial and Symmetric-Simplicial Groups

We extend the nonabelian Dold-Kan decomposition for simplicial groups of Carrasco and Cegarra in two ways. First, we show that the total order of the subgroups in their decomposition belongs to a family of total orders all giving rise to Dold-Kan decompositions. We exhibit a particular partial order such that the family is characterized as consisting of all total orders extending the partial order. Second, we consider symmetric-simplicial groups and show that, by using a specially chosen presentation of the category of symmetric-simplicial operators, new Dold-Kan decompositions exist which are algebraically much simpler than those of Carrasco and Cegarra in the sense that the commutator of two component subgroups lies in a single component subgroup.

math.AT↗

arXiv2010

On a Generalization of the Notion of Semidirect Product of Groups

We introduce an external version of the internal r-fold semidirect product of groups (SDP) of Carrasco and Cegarra. Just as for the classical external SDP, certain algebraic data are required to guarantee associativity of the construction. We give an algorithmic procedure for computing axioms characterizing these data. Additionally, we give criteria for determining when a family of homomorphisms from the factors of an SDP into a monoid or group assemble into a homomorphism on the entire SDP. These tools will be used elsewhere to give explicit algebraic axioms for hypercrossed complexes, which are algebraic models for classical homotopy types introduced by Carrasco and Cegarra.

math.AT↗