Abstract
? | Getzler and Kapronov introduced the notion of a modular operad to encode the
grafting of the stable algebraic curves along boundary components. These objects
have played a key role in the description of BV-algebra structures in
noncommutativity geometry and physics. Minor generalizations of this theory have
lead to the notion of compact symmetric multicategories which play a role in
categorical versions of quantum theory.
We wish to study a version of these objects where composition and contraction
are only defined up to (coherent) homotopy. To this end, we introduce an
appropriate category of undirected graphs $\mathfrak{M}$, and identify
subcategories of $\mathfrak{M}$-presheaves which model the indicated behavior.
For set-valued presheaves, these are objects satisfying an inner horn filling
condition. For space-valued presheaves these are objects satisfying a Segal-type
condition, which are also the fibrant objects in a certain Quillen model
structure. |
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