Injective stabilization in categories Alex Sorokin Northeastern - - PowerPoint PPT Presentation

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

• f 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 ΩA

⇁

⊗ ΣB − → A

⇁

⊗ B, where ΩA, ΣB 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 ΩA and a cosyzygy module ΣB; – Ker(A ⊗ η); – a map ΩA

⇁

⊗ ΣB − → 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 1C cyl 1C 1C

bot pr top

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 1C 1C path 1C

const start end

commute.

Alex Sorokin Injective stabilization in categories

Factorizations of diagonal and codiagonal

In a category C with binary coproducts any functorial factorization

• f codiagonal morphisms provides a cylinder

A ⊔ A cyl(A) A

(botA topA) ∇A prA

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 path(A) A ⊓ A

constA ∆A (startA endA)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 A cyl(A) A B

botA f H topA g

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 B path(B) B

H f g startB endB

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) A cone(A) ΣA

(1 0) baseA

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: ΩA path∗(A) A path(A) A ⊓ A

targetA (1 0)T

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 A cyl(A) B cyl(B)

botA f cyl(f ) botB

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 path(A) A path(B) B

startA path(f ) f startB

is a weak pull-back.

Alex Sorokin Injective stabilization in categories

Fibration and cofibration

Remark In a bicylindric category C the map A cone(A)

baseA

is a cofibration, i.e. the following diagram A cyl(A) cone(A) cyl(cone(A))

baseA botA cyl(baseA) botcone(A)

is a weak push-out.

Alex Sorokin Injective stabilization in categories

Fibration and cofibration

Remark In a bicylindric category C the map path∗(A) A

targetA

is a fibration, i.e. the following diagram path(path∗(A)) path∗(A) path(A) A

path(targetA) startpath∗(A) targetA startA

is a weak pull-back.

Alex Sorokin Injective stabilization in categories

Wedge sum of topological spaces

Wedge sum of pointed topological spaces (A, x0) and (B, y0) is A ∨ B = A ⊔ B {x0 = y0}

Alex Sorokin Injective stabilization in categories

Smash product of topological spaces

Smash product of pointed topological spaces (A, x0) and (B, y0) 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 cyl(A) = A ∧ I+, Σ(A) = A ∧ S1, path(A) = [I+, A], Ω(A) = [S1, A], where I+ is the interval [0, 1] with an adjoint base point, and S1 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 : I ⊔ I K I

∇I

= ⇒ A ⊔ A A ⊗ K A

∇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]

• n 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 ΩA

⇁

⊗ ΣB − → 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 . . . − → Ω2A

⇁

⊗ Σ2B − → ΩA

⇁

⊗ ΣB − → A

⇁

⊗ B.

Alex Sorokin Injective stabilization in categories