Higher Prop(erad)s. Philip Hackney, Marcy Robertson*, and Donald Yau UCLA. July 1, 2014

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1 Higher Prop(erad)s Philip Hackney, Marcy Robertson*, and Donald Yau UCLA July 1, 2014 Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

2 Intro: Enriched Categories A (small) category C consists of: a set of objects Ob(C); for each input-output pair (x, y), a set of arrows x f y so that: 1 every object x has an identity arrow x x and 2 there is an associative, unital composition given by contracting edges, i.e. f g x y z becomes x gf z id x Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

3 Intro: Enriched Categories Given a closed, symmetric monoidal category (V,, I), we define a V-enriched category C as: a set of objects Ob(C); for each input-output pair (x, y), an object C( y ) x in V so that: 1 every object x has an identity given by the map I id x C( x x ) and 2 there is an associative, unital composition given algebraically as a map in V. C( y x ) C(z y ) C( z x ) Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

4 Intro: Enriched Categories A map f C D consists of: a map of sets Ob(C) f 0 Ob(D); for each input-output pair (x, y), a morphism in V. C( y x ) f D( fy fx ) Moreover, we require that this data be compatible with composition and identities. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

5 Properads A properad P is a generalization of an category in which we allow morphisms to have a finite lists of objects for both the input and output, i.e. f (x 1,..., x m ) (y 1,..., y n ) with m, n 0. These (m, n)-operations are visualized as decorated corolla. y 1 x 1 f y n x m Remark: The objects of a properad, listed as (x 1,..., x m ; y 1,..., y n ) = (x; y), decorate the edges of the corolla. The (m, n)-operations decorate the vertices. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

6 Properads Properadic composition is represented by partially grafted corollas, i.e. g f Where grafting is defined when a non-empty sub-list of the outputs of f match a non-empty sub-list of the inputs of g. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

7 Example: Operads An operad is a special case of a properad in which a morphism has one output and finitely many inputs, i.e. (x 1,..., x n ) f y with n 0. We denote these n-operations by rooted trees. y x 1 f x n Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

8 Operads Composition of operations γ(f ; g 1,..., g n ) is represented by contracting edges of decorated (2-level) trees y f g 1 g n w1 1 wk 1 w n 1 1 wk n n to get decorated (1-level) trees y w 1 1 γ w n k n Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

9 Enriched Properads Given a closed, symmetric monoidal category (V,, I) we can define V-enriched properads. This means that we now decorate our corolla with objects P( y x ) V, i.e. y 1 x 1 P y n x m For each x Ob(P) we have an identity I P( x x ). Composition is written in terms of the tensor product of V, i P( w i x i ) P( x y ) γ P( w y ). Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

10 Enriched Properads A morphism f P Q of two V-enriched properads consists of: a map of sets f 0 Ob(P) Ob(Q); for each input-output list (x 1,..., x m ; y 1,..., y n ) = (x; y) a morphism in V. P( y x ) Q( f 0y f 0 x ) Moreover, all of this data is subject to compatibility with composition and identity maps. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

11 Model Category Structures The model category structure on Prop(V) blends together local and categorical information. We need the following facts: Definition The enriching category V is nice if V is a combinatorial monoidal model category with cofibrant unit which also satisfies Schwede-Shipley s monoid axiom. Definition For nice V, the connected components functor is defined as π 0 = Ho(V)(I, ) V Set Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

12 Model Category Structures Let Cat(V) denote the category of all small V-enriched categories.. Theorem (Muro, Berger-Moerdijk, Lurie) If V is nice, then Cat(V) admits a combinatorial model category structure where f C D is a weak equivalences (fibration) if Local For every input-output pair (x, y), the map C( y x ) D( f 0y f 0 x ) is a weak equivalence (fibration) in V. Categorical The induced map π 0 C π 0 D is an equivalence (fibration) of categories. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

13 Model Category Structures Let Prop(V) denote the category of all small V-enriched properads. Definition Let U 0 Prop(V) Cat(V) denote the functor which forgets all (m, n)-operations for m, n > 1. Theorem (Hackney-R) This functor has a left adjoint F 0 Cat(V) Prop(V) Definition For P Prop(V) the underlying category of P is given by π 0 (P) = π 0 U 0 (P). Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

14 Model Category Structures Theorem (Hackney-R-Yau) If V is nice, then Prop(V) admits a combinatorial model category structure where f P Q is a weak equivalences (fibration) if Local For every input-output list (x; y) the morphism Categorical The map P( y x ) Q( f 0y f 0 x ) is a weak equivalence (fibration) in V. π 0 P π 0 Q is an equivalence (fibration) of categories. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

15 Model Category Structures Example When V = sset, Prop(sSet) is a model for (, 1)-properads (already known in Hackney-Robertson II). Example If we restrict to the category of operads, Operad(V) has the model structure of Cisniski-Moerdijk (V = sset), Robertson (V = sset), or Caviglia (more general V). Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

16 Properness Theorem (Hackney-R-Yau) If the nice model structure on V is right proper, then so is the model category structure on Prop(V). Theorem (In progress) If all objects in the nice model structure on V are cofibrant, then the model category structure on Prop(V) is left proper. This can be restricted to get new information about operads Theorem (Hackney-R-Yau) If in the nice model structure on V is right proper, then so is the model category structure on Operad(V). If every object of V is cofibrant, then Operad(V) is left proper. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

17 Implications of Properness Theorem (In progress) There exists a Quillen equivalence between Prop(sSet) and the category of graphical sets, Set Γop Prop(sSet) hn Set Γop Today we focus on some new information about operads. Let Prop C (V) denote the category of V-enriched properads with C objects. Corollary (Hackney-R-Yau) Let V be a nice model category. If V is right proper, then so is the model structure on Operad C (V). If all objects in V are cofibrant, then Operad C (V) is left proper. hilip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

18 Change of Base Let V and W be two nice model categories. Definition (Schwede-Shipley) A Quillen pair between nice model categories V f W g is a weak monoidal Quillen adjunction if f g is a colax-lax monoidal adjunction so that if x and y are cofibrant objects in V the comultiplication f (x y) f (x) f (y) is a weak equivalence. A weak monoidal Quillen equivalence is a weak monoidal Quillen adjunction which is a Quillen equivalence. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

19 Change of Base We can extend the adjunction f g to an adjunction of properads. Theorem (Hackney-R-Yau) Assume V and W are nice model categories. Let V f W be a weak g monoidal Quillen adjunction. Then f g gives rise to a Quillen pair between model categories of properads, Prop(V) f prop Prop(W). g If f g is moreover an equivalence, then so is f prop g. a a Quillen equivalence requires additional assumption on V. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

20 Change of Base The adjunction is defined by: Prop(V) f prop Prop(W). g Applying the right adjoint g entry-wise, i.e. i g(p( w i )) g(p( x x i y )) lax g( i P( w i ) P( x x i y )). is defined by applying g(p( y )). x The lax monoidal structure implies that g commutes with the properad composition maps. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

21 Change of Base The left adjoint f prop exists for abstract reasons. If f is a strong symmetric monoidal functor, then f = f prop. We can make precise how close f and f prop are. Let P be a V-properad. There exists a map f (P( y x )) χ f prop (P( y x )) for each input-output list (x; y). Theorem If P is entry-wise cofibrant in V then χ P( y ) is a weak equivalence in W. x Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

22 Applications: Change of Base Corollary Extends Dold-Kan to symmetric operads and properads. There is a Quillen equivalence Prop(Mod(k) op ) Prop(Ch(k) 0 ) Can be used to extend Fresse s Rational Homotopy Type of operads to colored operads, properads. Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

23 Idea of the Proofs Both left properness and change of base are studyed by careful analysis of pushouts of properads A O B 1 Construct the Donor category, B objects of B are decorated, connected wheel-free graphs. we give an explicit partition of the vertices of any object of B into three distinct types: A, B, O We create full subcategories of B which are filtered based on how many instances of vertices of type A and O are allowed. 2 We take pushouts of the underlying graphs of the properads A,O, B whenever O A is a(n acyclic) cofibration. 3 This is realized a colimit of the pushouts over the filtered subcategories of B. 4 We show that this pushout can be lifted to a pushout of properads. hilip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1

24 References: 1 Philip Hackney and Marcy Robertson On the category of props 2 Philip Hackney and Marcy Robertson The homotopy theory of simplicial props 3 Philip Hackney, Marcy Robertson, Donald Yau Infinity properads and infinity wheeled properads (monograph) 4 Philip Hackney, Marcy Robertson, Donald Yau Left properness of colored operads (draft via request) 5 Marcy Robertson, Schematic homotopy types of operads (in progress) 6 Philip Hackney, Marcy Robertson, Donald Yau, Enriched properads and change of base (draft via request) 7 Fernando Muro, Homotopy theory of non-symmetric operads II: Change of base and left properness 8 Philip Hackney, Marcy Robertson, Donald Yau Homotopy coherent nerves of properads and A -categories (in progress) Philip Hackney, Marcy Robertson*, and Donald Yau (UCLA) Higher Prop(erad)s July 1, / 1