Diagrams encoding group actions

Transcription

1 University of California, Riverside July 27, 2012

2 Segal monoids Definition A Segal monoid is a functor X : op ssets such that X 0 = [0] and the Segal maps X n (X 1 ) n are weak equivalences for n 2. There is a model structure SeMon on the category of functors X : op ssets with X 0 = [0], with fibrant objects Segal monoids.

3 Segal monoids Definition A Segal monoid is a functor X : op ssets such that X 0 = [0] and the Segal maps X n (X 1 ) n are weak equivalences for n 2. There is a model structure SeMon on the category of functors X : op ssets with X 0 = [0], with fibrant objects Segal monoids. Theorem (B) There is a Quillen equivalence smon SeMon. In other words, every Segal monoid can be rigidified to a simplicial monoid.

4 Segal categories Definition A Segal category is a functor X : op ssets such that X 0 is discrete and the Segal maps are weak equivalences for n 2. X n X 1 X0 X0 }{{} n There is a model structure SeCat with fibrant objects Segal categories, which we can compare to categories enriched in simplicial sets, which form a model category SC.

5 Comparison to simplicial categories Theorem (B) There is a Quillen equivalence SC SeCat. These give two of the models for (, 1)-categories.

6 Segal groups To model a group structure, we can change the Segal maps. The usual Segal maps encode composition: ( ) [2]. On the other hand, Bousfield-Segal maps also force the existence of inverses: ( ) [2]. Theorem (Bousfield, B) There is a Quillen equivalence sgp SeGp.

7 Category actions on categories Definition The action of a category C on a category D, denoted C D, is given by: such that a moment map µ: Mor(D) Ob(C), an action map : Mor(C) s µ Mor(D) Mor(D), the moment respects the action: µ(f g) = t(f ), the action is associative, the identity map of any object acts trivially, µ(g g ) = µ(g), and s(f g) = s(g).

8 A morphism of actions is given by functors C C and D D respecting the moment and action maps. Category actions and morphisms form a category CatAct. Notice that it suffices to define the moment map only on objects, since µ(g) = µ(id t(g) g) = µ(id t(g) ).

9 Key example Example Let C 1,1 and D 1,1 be the categories 1 y z p p g p id x g x 0 w with moment map µ defined by µ(w) =, µ(x) = 0, µ(y) = µ(z) = 1. Then C 1,1 acts on D 1,1 as indicated by the maps in D 1,1.

10 Key example Example Let C 1,1 and D 1,1 be the categories 1 y z p p g p id x g x 0 w with moment map µ defined by µ(w) =, µ(x) = 0, µ(y) = µ(z) = 1. Then C 1,1 acts on D 1,1 as indicated by the maps in D 1,1. More generally, if A acts on B, then Mor(A) s µ Mor(B) = Hom CatAct (C 1,1 D 1,1, A B).

11 C n,k and D n,k More generally, define C n,k to be the category 0 k n. Define D n,k inductively, beginning with D 0,k = [k]. The moment map to the objects of C 0,k is given by { i 0 i < k µ(i) = 0 i = k.

12 C n,k and D n,k Then D 1,k looks like 0 1 k with µ( ) = k for each object.

13 C n,k and D n,k Then D 1,k looks like 0 1 k with µ( ) = k for each object. Then D n,k is built from D n 1,k by adding new objects (n, k) for each morphism h in D n 1,k with moment n 1, such that µ(n, k) = n, and morphisms p n 1,n h : s(h) (n, k).

14 Examples Figure: The categories D 0,2, D 1,2, and D 2,2

15 Action of C n,k on D n,k The action of C n,k on D n,k is given by p l,l h = p l l,l ( (p l,l+1 h)). Write [n k] for C n,k D n,k.

16 Action of C n,k on D n,k The action of C n,k on D n,k is given by p l,l h = p l l,l ( (p l,l+1 h)). Write [n k] for C n,k D n,k. A map X : [n k] A B is essentially a pair of functors α: [n] A and β : [k] B with µ(β(k)) = α(0). Taking all values of n and k gives a category.

17 The category Ω Definition (Moerdijk-Weiss) Let Ω be the category whose objects are rooted trees and whose morphisms are given by the operad morphisms between the (colored) operads generated by them. Denote by η the tree with no vertices and one edge. Definition A dendroidal set is a functor Ω op Sets. Given an object T of Ω, let Ω[T ] denote the representable functor Hom Ω (, T ), called the standard T -dendrex.

18 Dendroidal spaces Similarly, a dendroidal space is a functor X : Ω op ssets. Consider the category of dendroidal spaces X such that X (η) = [0]. We can also impose a condition similar to the Segal condition for simplicial spaces.

19 Dendroidal spaces Similarly, a dendroidal space is a functor X : Ω op ssets. Consider the category of dendroidal spaces X such that X (η) = [0]. We can also impose a condition similar to the Segal condition for simplicial spaces. Definition Let T be an object of Ω. Its Segal core Sc[T ] is the subobject of the T -dendrex given by all the corollas in T. In other words, Sc[T ] = v Ω[C n(v) ] where v denotes a vertex of T, n(v) denotes the number of input vertices of v, and C n denotes the n-corolla.

20 Segal operads So we have an inclusion Sc[T ] Ω[T ] for any tree T which we can use to define Segal maps. Definition A Segal operad is a dendroidal space X : Ω op ssets such that X (η) = [0] and the induced map Map(Ω[T ], X ) Map(Sc[T ], X ) is a weak equivalence of simplicial sets for all objects T of Ω. Since we have imposed the condition that X (η) = [0], we expect that Segal operads should correspond to ordinary (single-colored) operads.

21 Rigidification of Segal operads Theorem (B-Hackney) There is a model structure SeOper on the category of dendroidal spaces X with X (η) = [0] such that the fibrant objects are Segal operads.

22 Rigidification of Segal operads Theorem (B-Hackney) There is a model structure SeOper on the category of dendroidal spaces X with X (η) = [0] such that the fibrant objects are Segal operads. Let soper denote the category of simplicial operads. Theorem (B-Hackney) There is a Quillen equivalence soper SeOper. The Quillen equivalence between colored Segal operads and simplicial colored operads (without fixing the color) has been established by Cisinski and Moerdijk.

23 Categories acting on operads Let S be an object of Ω. Let C n,s be the free category with objects given by the edges of S and 0, 1,..., n, with 0 identified with the root edge of S, and morphisms generated by the p i,i+1 : i i + 1.

24 Categories acting on operads Let S be an object of Ω. Let C n,s be the free category with objects given by the edges of S and 0, 1,..., n, with 0 identified with the root edge of S, and morphisms generated by the p i,i+1 : i i + 1. Let O 0,S be the free operad on S. Define O n,s inductively similarly to the case for categories.

25 Categories acting on operads Let [n S] = C n,s O n,s. A map [n S] A P is determined by maps α: [n] A and β : S P. We get a category Ω of all such category actions.

26 Homotopy group actions on operads Consider the category of functors ( Ω) op ssets such that the image of [0 η] is [0]. We can localize with respect to the Segal maps for Ω and the Bousfield-Segal maps for to get a model structure Se(Gp Oper). Let sgp soper be the category of simplicial groups acting on simplicial operads.

27 Homotopy group actions on operads Consider the category of functors ( Ω) op ssets such that the image of [0 η] is [0]. We can localize with respect to the Segal maps for Ω and the Bousfield-Segal maps for to get a model structure Se(Gp Oper). Let sgp soper be the category of simplicial groups acting on simplicial operads. Theorem (B-Hackney) There is a Quillen equivalence sgp soper Se(Gp Oper).