Resumen de Combinatòria del pletisme via grupoides de segal i òperades

Alex Cebrian Galan

• The present thesis lies on the intersection between combinatorics, category theory and algebraic topology. We study the combinatorics of plethysm from the perspective of incidence bialgebras and objective combinatorics. The objective algebra is carried out at the level of Segal groupoids, by using homotopy slices and homotopy pullbacks of groupoids and simplicial methods.

  The first main contribution is to exhibit plethystic substitution as a convolution tensor product obtained from an explicit simplicial groupoid, TS, by the standard general constructions of incidence coalgebras and homotopy cardinality, in analogy with how ordinary substitution is obtained from the fat nerve NS of the category of finite sets and surjections S. The simplicial groupoid TS arises from S as its T-construction, a new categorical construction which is reminiscent of Quillen’s Q and Waldhausen’s S-constructions.

  The simplicial groupoid NS is equivalent to the two-sided bar construction of the operad Sym. We observe that TS too is equivalent to the two-sided bar construction of a certain operad, and that the way to obtain this operad from Sym can be generalized to any (nice enough) operad.

  This leads to the second main contribution: a functorial construction on generalized operads which formalizes the passage from ordinary substitutions to plethystic substitutions. This construction allows for treating simultaneously a variety of notions of plethysm, and in fact leads to new notions of plethysm. For all these notions of plethysm, a combinatorial model is exhibited in the form of a monoidal Segal groupoid.