Tensor topology Pau Enrique Moliner Chris Heunen Sean Tull 1 / 16 - PowerPoint PPT Presentation

Tensor topology Pau Enrique Moliner Chris Heunen Sean Tull 1 / 16

“Where things happen” ◮ Any monoidal category comes with built-in ‘space’ ◮ Matches examples ◮ Universal notion of support ◮ Completion to actual space ◮ Embedding separates out spatial dimension ◮ Coproducts correspond to complements See also [Balmer, “Tensor triangular geometry”] [Boyarchenko&Drinfeld, “Character sheaves of unipotent groups”] 2 / 16

Monoidal categories ◮ Objects ( A, B, C, . . . ) and morphisms ( f : A → B, . . . ) ◮ Two ways to compose: sequential ( ◦ ) and parallel ( ⊗ ) ◮ Two ways to do nothing: id A : A → A and I f ◦ id = id = id ◦ f I ⊗ A ≃ A ≃ A ⊗ I B B A D g f A C h k A A Morphisms I → I form commutative monoid of scalars Many examples: ◮ Hilbert spaces ◮ Sets ◮ Lattices 3 / 16

Idempotent subunits Categorify central idempotents in ring: � � ISub( C ) = s : S ֌ I | S ⊗ s : S ⊗ S → S ⊗ I iso 4 / 16

Example: order theory Frame: complete lattice, ∧ distributes over � e.g. open subsets of topological space 5 / 16

Example: order theory Frame: complete lattice, ∧ distributes over � e.g. open subsets of topological space Quantale: complete lattice, · distributes over � e.g. [0 , ∞ ], Pow( M ) 5 / 16

Example: order theory Frame: complete lattice, ∧ distributes over � e.g. open subsets of topological space Quantale: complete lattice, · distributes over � e.g. [0 , ∞ ], Pow( M ) ⊥ Quantale Frame ISub { x ∈ Q | x 2 = x ≤ 1 } Q 5 / 16

Example: logic ISub(Sh( X )) = { S ֌ 1 } = { S ⊆ X | S open } ∈ Frame 6 / 16

Example: algebra ISub( Mod R ) = � � � � S = S 2 = { x 1 y 1 + · · · + x n y n | x i , y i ∈ S } S ⊆ R ideal for nonunital bialgebra R in monoidal category 7 / 16

Example: analysis Hilbert module is C 0 ( X )-module with C 0 ( X )-valued inner product C 0 ( X ) = { f : X → C | ∀ ε > 0 ∃ K ⊆ X : | f ( X \ K ) | < ε } ISub( Hilb C 0 ( X ) ) = { S ⊆ X open } 8 / 16

Semilattice Proposition: ISub( C ) is a semilattice, ∧ = ⊗ , 1 = I T t I s S Caveat: C must be firm, i.e. s ⊗ T monic, and size issue 9 / 16

Semilattice Proposition: ISub( C ) is a semilattice, ∧ = ⊗ , 1 = I T t I s S Caveat: C must be firm, i.e. s ⊗ T monic, and size issue id ⊥ SemiLat FirmCat ISub 9 / 16

Spatial categories Call C spatial when ISub( C ) is frame ⊥ SemiLat Frame ⊣ ISub ⊣ ISub ? SpatCat FirmCat ⊥ 10 / 16

Spatial categories Call C spatial when ISub( C ) is frame ⊥ SemiLat Frame ⊣ ISub ⊣ ISub ? SpatCat FirmCat ⊥ Idea: � C = [ C op , Set ] is cocomplete 10 / 16

Spatial categories Call C spatial when ISub( C ) is frame ⊥ SemiLat Frame ⊣ ISub ⊣ ISub ? SpatCat FirmCat ⊥ Idea: � C = [ C op , Set ] is cocomplete � B,C C ( A, B ⊗ C ) × F ( B ) × G ( C ) F � ⊗ G ( A ) = Lemma : ISub( � C , � ⊗ ) is frame 10 / 16

Spatial categories Call C spatial when ISub( C ) is frame ⊥ SemiLat Frame ⊣ ISub ⊣ ISub ? SpatCat FirmCat ⊥ Idea: � C = [ C op , Set ] is cocomplete � B,C C ( A, B ⊗ C ) × F ( B ) × G ( C ) F � ⊗ G ( A ) = Lemma : ISub( � ⊗ ) is frame, but ISub( � � C , � C ) � = ISub( C ) 10 / 16

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I B ⊗ s 11 / 16

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I B ⊗ s f { s | s supports f } supp Pow(ISub( C )) C 2 11 / 16

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I B ⊗ s Monoidal functor: supp( f ) ∧ supp( g ) ≤ supp( f ⊗ g ) f { s | s supports f } supp Pow(ISub( C )) C 2 11 / 16

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I B ⊗ s Monoidal functor: supp( f ) ∧ supp( g ) ≤ supp( f ⊗ g ) f { s | s supports f } supp Pow(ISub( C )) C 2 � F F Q ∈ Frame universal with F ( f ) = � { F ( s ) | s ∈ ISub( C ) supports f } 11 / 16

Spatial completion Call F : C op → Set broad when � � F ( A ) ≃ ( f, s ): A → B | s ∈ supp( f ) ∩ U for some B ∈ C and U ⊆ ISub( C ). ⊥ SemiLat Frame ⊣ ISub ⊣ ISub SpatCat FirmCat ⊥ ˆ C C brd � = Sh( C , J )! 12 / 16

Simple categories Say C is simple when ISub( C ) = { id I } 13 / 16

Simple categories Say C is simple when ISub( C ) = { id I } universal property of localisation for Σ s = { A ⊗ s | A ∈ C } � ( − ) ⊗ S � s = C [Σ − 1 C s ] C ≃ F inverting Σ s D 13 / 16

Simple categories Say C is simple when ISub( C ) = { id I } universal property of localisation for Σ s = { A ⊗ s | A ∈ C } � ( − ) ⊗ S � s = C [Σ − 1 C s ] C ≃ F inverting Σ s D Lemma : Σ = { A ⊗ s | A ∈ C , s ∈ ISub( C } calculus of right fractions gives functor C → Loc( C ) = C [Σ − 1 ] into simple category 13 / 16

Slim categories Say C is slim when any object is (domain of) idempotent subunit (Note: S determines s ) Definition : support structure is functor ζ : C → C with morphisms ◮ β A : ζ ( A ) ֌ I ; ◮ γ A : A → ζ ( A ) ⊗ A ; ◮ δ A : ζ ( ζ ( A )) → ζ ( A ); satisfying five coherence conditions Example : supported quantales Proposition : δ A is iso, β A is idempotent, ζ : C → ISub( C ) 14 / 16

Slim categories Say C is slim when any object is (domain of) idempotent subunit (Note: S determines s ) Definition : support structure is functor ζ : C → C with morphisms ◮ β A : ζ ( A ) ֌ I ; ◮ γ A : A → ζ ( A ) ⊗ A ; ◮ δ A : ζ ( ζ ( A )) → ζ ( A ); satisfying five coherence conditions Example : supported quantales Proposition : δ A is iso, β A is idempotent, ζ : C → ISub( C ) Theorem : Any supported monoidal category embeds into product of simple and slim one: C ֌ Loc( C ) × ISub( C ) 14 / 16

Complements s Subunit is split when id S I SISub( C ) is a sub-semilattice of ISub( C ) (don’t need firmness) 15 / 16

Complements s Subunit is split when id S I SISub( C ) is a sub-semilattice of ISub( C ) (don’t need firmness) If C has zero object, ISub( C ) has least element 0 s, s ⊥ are complements if s ∧ s ⊥ = 0 and s ∨ s ⊥ = 1 15 / 16

Complements s Subunit is split when id S I SISub( C ) is a sub-semilattice of ISub( C ) (don’t need firmness) If C has zero object, ISub( C ) has least element 0 s, s ⊥ are complements if s ∧ s ⊥ = 0 and s ∨ s ⊥ = 1 Proposition : when C has finite biproducts, then s, s ⊥ ∈ SISub( C ) are complements if and only if they are biproduct injections Corollary : if ⊕ distributes over ⊗ , then SISub( C ) is a Boolean algebra (universal property?) 15 / 16

Conclusion ◮ Any monoidal category comes with built-in ‘space’ ◮ Matches examples ◮ Universal notion of support ◮ Completion to actual space ◮ Embedding separates out spatial dimension ◮ Coproducts correspond to complements Further goals: ◮ Canonical status for support structure ◮ Dauns-Hofmann-like theorem ◮ Graphical calculus ◮ Applications: causality, concurrency 16 / 16

Restriction � � The full subcategory C s of A with A ⊗ s invertible is: ◮ monoidal with tensor unit S � � ◮ coreflective: C ⊥ C s � � � � ◮ tensor ideal: if A ∈ C and B ∈ C s , then A ⊗ B ∈ C s � � ◮ monocoreflective: counit ε I monic (and A ⊗ ε I iso for A ∈ C s )

Restriction � � The full subcategory C s of A with A ⊗ s invertible is: ◮ monoidal with tensor unit S � � ◮ coreflective: C ⊥ C s � � � � ◮ tensor ideal: if A ∈ C and B ∈ C s , then A ⊗ B ∈ C s � � ◮ monocoreflective: counit ε I monic (and A ⊗ ε I iso for A ∈ C s ) Proposition : ISub( C ) ≃ { monocoreflective tensor ideals in C }

Restriction � � The full subcategory C s of A with A ⊗ s invertible is: ◮ monoidal with tensor unit S � � ◮ coreflective: C ⊥ C s � � � � ◮ tensor ideal: if A ∈ C and B ∈ C s , then A ⊗ B ∈ C s � � ◮ monocoreflective: counit ε I monic (and A ⊗ ε I iso for A ∈ C s ) Proposition : ISub( C ) ≃ { monocoreflective tensor ideals in C } � � � � Examples: ( Mod R ) I = Mod I , Sh( X ) U = Sh( U )

Localisation A graded monad is a monoidal functor E → [ C , C ] ( η : A → T (1), µ : T ( t ) ◦ T ( s ) → T ( s ⊗ t )) � � Lemma : s �→ C s is an ISub( C )-graded monad

Localisation A graded monad is a monoidal functor E → [ C , C ] ( η : A → T (1), µ : T ( t ) ◦ T ( s ) → T ( s ⊗ t )) � � Lemma : s �→ C s is an ISub( C )-graded monad universal property of localisation for Σ s = { A ⊗ s | A ∈ C } ( − ) ⊗ S � � s = C [Σ − 1 C s ] C ≃ F inverting Σ s D

Localisation A graded monad is a monoidal functor E → [ C , C ] ( η : A → T (1), µ : T ( t ) ◦ T ( s ) → T ( s ⊗ t )) � � Lemma : s �→ C s is an ISub( C )-graded monad universal property of localisation for Σ s = { A ⊗ s | A ∈ C } ( − ) ⊗ S � � s = C [Σ − 1 C s ] C ≃ F inverting Σ s D Lemma : Σ = { A ⊗ s | A ∈ C , s ∈ ISub( C } calculus of right fractions gives functor C → Loc( C ) = C [Σ − 1 ] into simple category