A graphical category for higher modular operads

Abstract

We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted U, plays a similar role for modular operads that the dendroidal category Ω plays for operads. We carefully study properties of U, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from U.A modular operad, as introduced by Getzler and Kapranov [GK98], is a kind of cyclic operad equipped with self-compositions of operations. That is, it is an algebraic structure consisting of a sequence of sets P (n), indexed on nonnegative integers n, together with families of 'composition operations' P (n) × P (m) → P (n + m − 2) and 'contraction operations' P (n) → P (n − 2). The canonical example is when P (n) is the moduli space of Riemann surfaces with n marked points. This paper, along with its companion [HRY], center around a new category of graphs that permit a Segalic approach to the study of modular operads.The main goal of the present paper is to propose a precise definition for upto-homotopy modular operads and provide a homotopy theory for such objects. Why might one pursue such a program? One motivation comes from the work of Mann and Robalo who show, by passing to correspondences in derived stacks, that the operad of stable curves of genus zero acts on any smooth projective complex variety (see [MR18, Theorem 1.1.2]). This allows them to construct (genus zero) Gromov-Witten invariants on the derived category of the variety in question. We anticipate this work will be used in the study of higher genus version of that result; see Remark 1.2.1 of [MR18], which is prefigured in the ordinary case in [Bar07, 11.2].An additional motivation comes from work of the second author with Horel and Boavida de Brito on subgroups of the profinite Grothendieck-Teichmüller group. They conjecture that the homotopy automorphisms of a profinite completion of Getzler-Kapronov's modular operad M

Brauer diagrams, modular operads, and a graphical nerve theorem for circuit algebras

“…When V = Set, it is possible, by enlarging Σ, to describe all coloured collections (A(c)) c∈listC (for any palette (C, ω)) underlying circuit algebras in CA, as presheaves on the same category. This is the idea of graphical species, that were introduced in [16], and used in the definition of coloured Set-valued modular operads (compact closed categories) [16,30,13,14]. This section provides a short discussion on graphical species in an arbitrary category E with finite limits.…”

“…The remainder of the present paper is concerned with an analogous construction for circuit operads. This section reviews the definition of the category Gr et of graphs and étale morphisms -introduced in [16], and used in [21,13,14,30] -and describes how the graphs in this category are related to wiring diagrams. The interested reader is referred to [30,Sections 3 & 4] for more details and explicit proofs related to the graphical formalism.…”

“…This section is largely a review of basic graph constructions. As in [30] (and [13,14]), this is based on the definition of graphs introduced in [16].…”

Dwyer–Kan homotopy theory for cyclic operads

“…In analogy with the Moerdijk-Weiss category Ω from [34], the category of positive cyclic operads admits a full subcategory Ω Σ cyc of unrooted trees. The non-symmetric version (for planar unrooted trees), Ω cyc , is explained in the work of Tashi Walde [41], while the symmetric version appears in work of Hackney, Robertson, and Yau [23]. The category Ω Σ cyc admits a bijective-onobjects full functor to the category Ξ from [22].…”

Modular operads and the nerve theorem

“…Remark 3.10. If Y is a Segal U-presheaf, then its right Kan extension ι * Y along the inclusion ι : U → U is a Segal U-presheaf [HRY,Theorem 4.11]. By definition, the modular operad associated to Y is just the modular operad associated to the Segal U-presheaf ι * Y (Definition 3.17).…”