49 research outputs found
Categorified cyclic operads
In this paper, we introduce a notion of categorified cyclic operad for
set-based cyclic operads with symmetries. Our categorification is obtained by
relaxing defining axioms of cyclic operads to isomorphisms and by formulating
coherence conditions for these isomorphisms. The coherence theorem that we
prove has the form "all diagrams of canonical isomorphisms commute". Our
coherence results come in two flavours, corresponding to the "entries-only" and
"exchangeable-output" definitions of cyclic operads. Our proof of coherence in
the entries-only style is of syntactic nature and relies on the coherence of
categorified non-symmetric operads established by Do\v{s}en and Petri\'c. We
obtain the coherence in the exchangeable-output style by "lifting" the
equivalence between entries-only and exchangeable-output cyclic operads, set up
by the second author. Finally, we show that a generalisation of the structure
of profunctors of B\' enabou provides an example of categorified cyclic operad,
and we exploit the coherence of categorified cyclic operads in proving that the
Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd
version.Comment: 57 page
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio