The Homotopy Theory of Simplicially Enriched Multicategories

THE HOMOTOPY THEORY OF SIMPLICIALLY ENRICHED MULTICATEGORIES MARCY ROBERTSON Abstract. In this paper we construct a cofibrantly generated model category structure on the category of all small symmetric multicategories enriched in simplicial sets. Operads are combinatorial objects that encode a variety of algebraic structures in a particular symmetric monoidal category of interest. Many usual categories of algebras (i.e. categories of commutative and associa- tive algebras, associative algebras, Lie algebras, Poisson algebras, etc.) can be considered categories of operad representations. At the same time, operads are “algebras” themselves, or rather monoids in the monoidal category of (symmetric) sequences. A multicategory, or colored operad, is simply an operad with “many objects,” analogous to the way a category is a monoid with “many objects.” The precise definition and some important examples of operads, multicategories and their algebras will be reviewed in Section 1 of this paper. The purpose of this paper is the construction of a Quillen model category structure on the category of small multicategories enriched in simplicial sets. Our model structure is a blending of the Bergner model structure on the category of small simplicial categories [Be], and the Berger-Moerdijk model structure on S-colored operads [BM07]. Acknowledgments: The model category structure presented here for simplicially enriched symmetric mul- ticategories was independently obtained by the author as part of her thesis [Robertson] work and Ieke Moerdijk [M] as part of a larger project on (∞, 1)-operads [CM09, CM10, CM]. The author is greatly in- debted to her thesis advisor, Brooke Shipley, and to Ieke Moerdijk for the many helpful discussions, and to the later for showing her his unpublished manuscript of which she has made liberal use in the preparation of this paper. 1. Multicategories The basic idea of a multicategory is very like the idea of a category, it has objects and morphisms, but in a multicategory the source of a morphism can be an arbitrary sequence of objects rather than just a single object. A multicategory, P, consists of the following data: • a set of objects obj(P); • for each n ≥0 and each sequence of objects x1, ..., xn, x a set P(x1, ..., xn; x) of n-ary operations which take n inputs (the sequence x1, ..., xn) to a single output (the object x). These operations are equipped with structure maps for units and composition. Specifically, if I = {∗} denotes the one-point set, then for each object x there exists a unit map ηx : I →P(x; x) taking ∗to Key words and phrases. Colored operad; multicategory. 1 arXiv:1111.4146v1 [math.AT] 17 Nov 2011 1x, where 1 denotes the unit of the symmetric monoidal structure on the category Set. The composition operations are given by maps P(x1, ..., xn; x) × P(y1 1, ..., y1 k1; x1) × · · · × P(yn 1 , ..., yn kn; xn) −→P(y1 1, ..., yn kn; x) which we denote by p, q1, ..., qn 7→p(q1, ..., qn). The structure maps satisfy the associativity and unitary coherence conditions of monoids. A symmetric multicategory is a multicategory with the additional property that the n-ary operations are equivariant under the permutation of the inputs. Explicitly, for σ ∈Σn and each sequence of objects x1, ..., xn, x we have a right action of Σn, i.e., a morphism σ∗: P(x1, · · · , xn; x) −→P(xσ(1), ..., xσ(n); x). The action maps are well behaved, in the sense that all composition operations are invariant under the Σn-actions, and (στ)∗= τ ∗σ∗. In practice one often uses the following, equivalent, definition of the composition operations, given by: P(c1, · · · , cn; c) × P(d1, · · · , dk; ci) ◦i / P(c1, · · · , ci−1, d1, · · · , dk, ci+1, · · · , cn; c). All of our definitions will still make sense if we replace Set by any co-complete symmetric monoidal cat- egory (C, ⊗, 1). Multicategories whose operations take values in C are called multicategories enriched in C or C-multicategories. In particular, the strong monoidal functor Set −→C that sends a set S to the S-fold coproduct of copies of the unit of C takes every multicategory to a C-enriched multicategory. Example 1 (Enriched Categories). Let S be a set and let (C, ⊗, 1) be a symmetric monoidal category. There exists a (non-symmetric) S-colored operad CatS whose algebras are the C-enriched categories with S as set of objects and where the maps between algebras (i.e. functors between the C-enriched categories with object set S) are the functors which act by the identity on objects. One puts CatS((x1, x′ 1), . . . , (xn, x′ n); (x′ 0, xn+1)) = 1 whenever x′ i = xi+1 for i = 0, . . . , n, and zero in all other cases. In particular, for n = 0 we have CatS(; (x, x)) = 1 for each x ∈S, providing the CatS-algebras with the necessary identity arrows. Example 2 (Operad Homorphisms). Let P be an arbitrary operad. There exists a colored operad P1 on a set {0, 1} of two colors, whose algebras are triples (A0, A1, f) where A0 and A1 are P-algebras, and f : A0 →A1 is

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