Co-rings Over Operads preliminaries Diffraction and cobar duality - PowerPoint PPT Presentation

Co-rings Over Operads Kathryn Hess Motivating example Category- theoretic Co-rings Over Operads preliminaries Diffraction and cobar duality Enriched Kathryn Hess induction Future work Institute of Geometry, Algebra and Topology Appendix: bundles of Ecole Polytechnique Fédérale de Lausanne bicategories with connection CT 2006, White Point, Nova Scotia, 27 June 2006

Co-rings Over Slogan Operads Kathryn Hess Motivating Operads example Category- parametrize n -ary operations, and theoretic preliminaries govern the identities that they must satisfy. Diffraction and cobar duality Enriched induction Future work Appendix: bundles of bicategories with connection

Co-rings Over Slogan Operads Kathryn Hess Motivating Operads example Category- parametrize n -ary operations, and theoretic preliminaries govern the identities that they must satisfy. Diffraction and cobar duality Enriched Co-rings over operads induction Future work parametrize higher, “up to homotopy” structure on Appendix: homomorphisms, and bundles of bicategories with govern the relations among the “higher homotopies" connection and the n -ary operations.

Co-rings Over Slogan Operads Kathryn Hess Motivating Operads example Category- parametrize n -ary operations, and theoretic preliminaries govern the identities that they must satisfy. Diffraction and cobar duality Enriched Co-rings over operads induction Future work parametrize higher, “up to homotopy” structure on Appendix: homomorphisms, and bundles of bicategories with govern the relations among the “higher homotopies" connection and the n -ary operations. Co-rings over operads should therefore be considered as relative operads.

Co-rings Over Notation and conventions Operads Kathryn Hess Ch is the category of chain complexes over a Motivating commutative ring R that are bounded below. example Category- Ch is closed, symmetric monoidal with respect to the theoretic preliminaries tensor product: Diffraction and cobar duality ( C , d ) ⊗ ( C ′ , d ′ ) := ( C ′′ , d ′′ ) Enriched induction Future work where � C ′′ C i ⊗ R C ′ Appendix: n = j bundles of bicategories with i + j = n connection and d ′′ = d ⊗ R C ′ + C ⊗ R d ′ .

Co-rings Over Notation and conventions Operads Kathryn Hess Ch is the category of chain complexes over a Motivating commutative ring R that are bounded below. example Category- Ch is closed, symmetric monoidal with respect to the theoretic preliminaries tensor product: Diffraction and cobar duality ( C , d ) ⊗ ( C ′ , d ′ ) := ( C ′′ , d ′′ ) Enriched induction Future work where � C ′′ C i ⊗ R C ′ Appendix: n = j bundles of bicategories with i + j = n connection and d ′′ = d ⊗ R C ′ + C ⊗ R d ′ . (Co)monoids in a given monoidal category are not assumed to be (co)unital.

Co-rings Over Outline Operads Kathryn Hess Motivating example Motivating example 1 Category- theoretic preliminaries Category-theoretic preliminaries 2 Diffraction and cobar duality Co-rings Enriched Operads as monoids induction Future work Appendix: Diffraction and cobar duality 3 bundles of bicategories with connection Enriched induction 4 Appendix: bundles of bicategories with connection 5

Co-rings Over The cobar construction Operads Kathryn Hess Let C denote the category of chain coalgebras, i.e., of Motivating comonoids in Ch . Let A denote the category of chain example algebras, i.e., of monoids in Ch . Category- theoretic preliminaries The cobar construction is a functor Diffraction and � � cobar duality T ( s − 1 C ) , d Ω Ω : C − → A : C �− → Ω C = , Enriched induction where Future work Appendix: T is the free monoid functor on graded R -modules, bundles of bicategories with ( s − 1 C ) n = C n + 1 for all n , and connection d Ω is the derivation specified by d Ω s − 1 = − s − 1 d + ( s − 1 ⊗ s − 1 )∆ , where d and ∆ are the differential and coproduct on C .

Co-rings Over The category DCSH Operads Kathryn Hess Motivating example Category- theoretic preliminaries Ob DCSH = Ob C . Diffraction and DCSH ( C , C ′ ) := A (Ω C , Ω C ′ ) . cobar duality Enriched induction Future work Appendix: bundles of bicategories with connection

Co-rings Over The category DCSH Operads Kathryn Hess Motivating example Category- theoretic preliminaries Ob DCSH = Ob C . Diffraction and DCSH ( C , C ′ ) := A (Ω C , Ω C ′ ) . cobar duality Enriched Morphisms in DCSH are called strongly induction Future work homotopy-comultiplicative maps. Appendix: bundles of ⇒ { ϕ k : C → ( C ′ ) ⊗ k } k ≥ 1 + relations! ϕ ∈ DCSH ( C , C ′ ) ⇐ bicategories with connection

Co-rings Over The category DCSH Operads Kathryn Hess Motivating example Category- theoretic preliminaries Ob DCSH = Ob C . Diffraction and DCSH ( C , C ′ ) := A (Ω C , Ω C ′ ) . cobar duality Enriched Morphisms in DCSH are called strongly induction Future work homotopy-comultiplicative maps. Appendix: bundles of ⇒ { ϕ k : C → ( C ′ ) ⊗ k } k ≥ 1 + relations! ϕ ∈ DCSH ( C , C ′ ) ⇐ bicategories with connection The chain map ϕ 1 : C → C ′ is called a DCSH-map.

Co-rings Over Topological significance I Operads Kathryn Hess Motivating example Category- theoretic Let K be a simplicial set. preliminaries Diffraction and cobar duality Theorem (Gugenheim-Munkholm) Enriched induction The natural coproduct ∆ K : C ∗ K → C ∗ K ⊗ C ∗ K is Future work naturally a DCSH-map Appendix: bundles of � �� � bicategories with Thus, there exists ϕ K ∈ A Ω C ∗ K , Ω C ∗ K ⊗ C ∗ K such connection that ( ϕ K ) 1 = ∆ K .

Co-rings Over Topological significance II Operads Kathryn Hess Motivating Theorem (H.-Parent-Scott-Tonks) example Category- There is a natural, coassociative coproduct ψ K on Ω C ∗ K, theoretic preliminaries given by the composite Diffraction and cobar duality � � q ϕ K Ω C ∗ K − − → Ω C ∗ K ⊗ C ∗ K → Ω C ∗ K ⊗ Ω C ∗ K , − Enriched induction Future work where q is Milgram’s natural transformation. Appendix: bundles of Furthermore, Szczarba’s natural equivalence of chain bicategories with connection algebras ≃ Sz : Ω C ∗ K − → C ∗ GK is a DCSH-map with respect to ψ K and to the natural coproduct ∆ GK on C ∗ GK.

Co-rings Over Monoidal products of bimodules Operads Kathryn Hess Motivating example Let ( M , ⊗ , I ) be a bicomplete monoidal category. Let Category- theoretic ( A , µ ) be a monoid in M . preliminaries Co-rings Operads as monoids Diffraction and Remark cobar duality Enriched The category of A -bimodules is also monoidal, with induction monoidal product ⊗ A given by the coequalizer Future work Appendix: bundles of bicategories with ρ ⊗ N connection M ⊗ A ⊗ N M ⊗ N − → M ⊗ A N . ⇒ M ⊗ λ

Co-rings Over Definition of co-rings Operads Kathryn Hess Motivating example Category- theoretic An A -co-ring is a comonoid ( R , ψ ) in the category of preliminaries Co-rings A -bimodules, i.e., Operads as monoids Diffraction and cobar duality ψ : R − → R ⊗ A R Enriched induction is coassociative. Future work Appendix: bundles of CoRing A is the category of A -co-rings and their bicategories with connection morphisms.

� � � � Co-rings Over Example: the canonical co-ring Operads Kathryn Hess Motivating Let ϕ : B → A be a monoid morphism. example Category- Let R = A ⊗ B A . theoretic preliminaries Co-rings Define ψ : R → R ⊗ A R to be following composite of Operads as monoids Diffraction and A -bimodule maps. cobar duality Enriched induction ∼ A ⊗ B A = A ⊗ B B ⊗ B A Future work Appendix: A ⊗ ϕ ⊗ A bundles of ψ B B bicategories with connection ∼ ( A ⊗ B A ) ⊗ A ( A ⊗ B A ) = A ⊗ B A ⊗ B A This example arose in Galois theory.

Co-rings Over Kleisli constructions I Operads Kathryn Hess Motivating example ( A , R ) Mod Category- theoretic preliminaries Ob ( A , R ) Mod = Ob A Mod Co-rings Operads as monoids Diffraction and ( A , R ) Mod ( M , N ) = A Mod ( R ⊗ A M , N ) cobar duality Enriched induction Composition of ϕ ∈ ( A , R ) Mod ( M , M ′ ) and Future work ϕ ′ ∈ ( A , R ) Mod ( M ′ , M ′′ ) given by the composite of left Appendix: bundles of A -module morphisms below. bicategories with connection ψ ⊗ A M R ⊗ A ϕ ϕ ′ A M ′ → M ′′ R ⊗ A M − − − → R ⊗ A R ⊗ A M − − − → R ⊗ −

Co-rings Over Kleisli constructions II Operads Kathryn Hess Motivating example Mod ( A , R ) Category- theoretic preliminaries Ob Mod ( A , R ) = Ob Mod A Co-rings Operads as monoids Diffraction and Mod ( A , R ) ( M , N ) = Mod A ( M ⊗ A R , N ) cobar duality Enriched induction Composition of ϕ ∈ Mod ( A , R ) ( M , M ′ ) and Future work ϕ ′ ∈ Mod ( A , R ) ( M ′ , M ′′ ) given by the composite of Appendix: bundles of right A -module morphisms below. bicategories with connection M ⊗ A ψ ϕ ⊗ A R ϕ ′ → M ′ ⊗ → M ′′ M ⊗ A R − − − → M ⊗ A R ⊗ A R − − − A R −