Injective stabilization in categories Alex Sorokin Northeastern University, Boston P¨ arnu, July 17, 2019 Alex Sorokin Injective stabilization in categories
Motivation Martsinkovsky, Russel (2017): asymptotic stabilization of tensor product over an arbitrary associative ring R is obtained as a limit of the sequence of maps built by iterating the following steps: – given right and left R − modules A , B consider a map B I , where I is an injective R -module; – define an injective stabilization of A ⊗ B as ⇁ A ⊗ B = Ker( A ⊗ B − → A ⊗ I ) – construct a map ⇁ ⇁ Ω A ⊗ Σ B − → A ⊗ B , where Ω A , Σ B are the modules of syzygies and cosyzygies respectively. Alex Sorokin Injective stabilization in categories
Key ingredients For an asymptotic stabilization of A ⊗ − we need: – an embedding η of B in an injective module I ; – a syzygy module Ω A and a cosyzygy module Σ B ; – Ker( A ⊗ η ); ⇁ ⇁ – a map Ω A ⊗ Σ B − → A ⊗ B . Alex Sorokin Injective stabilization in categories
Question and goal Question : in what kind of categories with tensor product can be defined an asymptotic stabilization? Answer: a category C should have – all finite limits and colimits; – a zero object; – a bifunctor ⊗ , such that for each A the functor A ⊗ − has right adjoint; – a syzygy and cosyzygy endofunctors Ω and Σ on C . Question : What will play the role of short exact sequences? Answer: (Co)fibration sequences. Thus, we need some abstract homotopy theory in our category. Alex Sorokin Injective stabilization in categories
Cylinder and Cocylinder Definition For a category C a cylinder functor on C is an endofunctor cyl : C → C equipped with natural transformations bot , top , pr that make the following diagram top bot cyl 1 C 1 C pr 1 C commute. Alex Sorokin Injective stabilization in categories
Cylinder and Cocylinder Definition For a category C a cocylinder functor ( path space functor ) on C is an endofunctor path : C → C equipped with natural transformations start , end , const that make the following diagram 1 C const path 1 C 1 C start end commute. Alex Sorokin Injective stabilization in categories
Factorizations of diagonal and codiagonal In a category C with binary coproducts any functorial factorization of codiagonal morphisms provides a cylinder (bot A top A ) A ⊔ A cyl( A ) pr A ∇ A A and any cylinder arises in this way. Alex Sorokin Injective stabilization in categories
Factorizations of diagonal and codiagonal In a category C with binary products any functorial factorization of diagonal morphisms provides a cocylinder A ∆ A const A path( A ) A ⊓ A (start A end A ) T and any cocylinder arises in this way. Alex Sorokin Injective stabilization in categories
Left and right homotopies Definition Let C be a category with cylinder. Morphisms f , g : A → B are left homotopic if there exists a morphism H making the following diagram top A bot A cyl( A ) A A f H g B commute. Alex Sorokin Injective stabilization in categories
Left and right homotopies Definition Let C be a category with cocylinder. Morphisms f , g : A → B are right homotopic if there exists a morphism H making the following diagram A g f H B path( B ) B start B end B commute. Alex Sorokin Injective stabilization in categories
Left and right homotopies If in category C cyl ⊣ path then f , g are left homotopic ⇐ ⇒ f , g are right homotopic Alex Sorokin Injective stabilization in categories
Convinient setting Definition We will say that a category C is bicylindric if: – C is pointed (i.e. has zero object); – C is finitely bicomplete (i.e. has all finite limits and colimits); – C is equipped with an adjoint pair of cylinder-cocylinder functors. As the examples of such categories C we can consider – all pointed small categories Cat ∗ / ; – all pointed compactly generated topological spaces kTop ∗ / ; – all pointed sets Set ∗ / ; – the category of complexes Comp ( A ) over an abelian category A . Alex Sorokin Injective stabilization in categories
Suspension and loop space Definition In a bicylindrical category C a cone and a suspension endofunctors cone , Σ are defined by the following push-out diagrams: A ⊔ A cyl( A ) ( 1 0 ) base A cone( A ) A 0 Σ A Alex Sorokin Injective stabilization in categories
Suspension and loop space Definition In a bicylindrical category C a based path space and a loop space endofunctors path ∗ , Ω are defined by the following pull-back diagrams: Ω A 0 target A path ∗ ( A ) A ( 1 0 ) T path( A ) A ⊓ A Alex Sorokin Injective stabilization in categories
Fibrations and cofibrations Definition In a category C with a cylinder a morphism f : A → B is a cofibration if bot A A cyl( A ) f cyl( f ) bot B B cyl( B ) is a weak push-out. Alex Sorokin Injective stabilization in categories
Fibrations and cofibrations Definition In a category C with a cocylinder a morphism f : A → B is a fibration if start A path( A ) A path( f ) f start B path( B ) B is a weak pull-back. Alex Sorokin Injective stabilization in categories
Fibration and cofibration Remark In a bicylindric category C the map base A cone( A ) A is a cofibration, i.e. the following diagram bot A A cyl( A ) base A cyl(base A ) bot cone( A ) cone( A ) cyl(cone( A )) is a weak push-out. Alex Sorokin Injective stabilization in categories
Fibration and cofibration Remark In a bicylindric category C the map target A path ∗ ( A ) A is a fibration, i.e. the following diagram start path ∗ ( A ) path(path ∗ ( A )) path ∗ ( A ) path(target A ) target A start A path( A ) A is a weak pull-back. Alex Sorokin Injective stabilization in categories
Wedge sum of topological spaces Wedge sum of pointed topological spaces ( A , x 0 ) and ( B , y 0 ) is A ⊔ B A ∨ B = { x 0 = y 0 } Alex Sorokin Injective stabilization in categories
Smash product of topological spaces Smash product of pointed topological spaces ( A , x 0 ) and ( B , y 0 ) is A ∧ B = A × B A ∨ B Alex Sorokin Injective stabilization in categories
Smash product of topological spaces Remark In kTop ∗ / , using the smash product ∧ and the space of maps [ , ] , we can express the usual notions of cylinder, path space, suspension and cylinder as Σ( A ) = A ∧ S 1 , cyl( A ) = A ∧ I + , Ω( A ) = [ S 1 , A ] , path( A ) = [ I + , A ] , where I + is the interval [ 0 , 1 ] with an adjoint base point, and S 1 is a circle with a base point. Alex Sorokin Injective stabilization in categories
Tensor product and (co)cylinder Let ( C , ⊗ , [ , ] , I ) be a closed symmetric monoidal category. Then any factorization of the codiagonal ∇ I defines a cylinder on C : A ⊔ A A ⊗ K I ⊔ I K = ⇒ ∇ I ∇ A I A Alex Sorokin Injective stabilization in categories
Tensor product and (co)cylinder Any factorization of ∇ I defines a cylinder A �→ A ⊗ K and a cocylinder A �→ [ K , A ] on C . Remark Not every cylinder on C can be defined as A �→ A ⊗ K Alex Sorokin Injective stabilization in categories
Injective stabilization in dyadic category Definition A closed symmetric monoidal bicylindric category ( C , ⊗ , [ , ] , I ) , whose cylinder-cocylinder pair is induced by the tensor product, is said to be dyadic . Alex Sorokin Injective stabilization in categories
Injective stabilization in dyadic category Definition An injective stabilization of a functor A ⊗ − in a dyadic category C is ⇁ A ⊗ − = Ker( A ⊗ − − → A ⊗ cone( − )) . Alex Sorokin Injective stabilization in categories
Asymptotic stabilization Theorem In a dyadic category C there is a map ⇁ ⇁ Ω A ⊗ Σ B − → A ⊗ B , which is functorial in A an B . Alex Sorokin Injective stabilization in categories
Asymptotic stabilization Definition Let C be a complete dyadic category. An asymptotic stabilization T ( A , − ) of a functor A ⊗ − is a limit of a sequence ⇁ ⇁ ⇁ → Ω 2 A ⊗ Σ 2 B − . . . − → Ω A ⊗ Σ B − → A ⊗ B . Alex Sorokin Injective stabilization in categories
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