Some categorical perspectives on the modular envelope of a cyclic operad

Getzler and Kapranov defined the notion of a modular
operad in the 1990's. More recently Costello, motivated by the
problem of classifying TCFT's, gave an alternative perspective of
cyclic and modular operads, and expressed the modular envelope
of a cyclic operad as the process of left Kan extending along a
specific functor between categories whose morphisms are graphs of
some kind. Various aspects of this story were not originally made
precise, though some subsequent work, most notably by Getzler
and Kaufmann-Ward have filled in much of the story. In this talk,
I'll review these developments, and explain how Guitart's notion
of "exact square", and parts of 2-dimensional monad theory lead
to further clarifications. Part of this is joint work with Michael
Batanin and Joachim Kock.