Tensor topology Chris Heunen Pau Enrique Moliner Sean Tull 1 / 15 - PowerPoint PPT Presentation

Tensor topology Chris Heunen Pau Enrique Moliner Sean Tull 1 / 15

“Where things happen” Wouldn’t it be great if ◮ control flow ◮ data flow ◮ provenance ◮ proof analysis ◮ causality were all instances of a one theory? 2 / 15

Idempotent subunits Categorify idempotents in ring � � ISub( C ) = s : S ֌ I | id S ⊗ s : S ⊗ S → S ⊗ I iso 3 / 15

Example: order theory Frame: complete lattice, ∧ distributes over � e.g. open subsets of topological space 4 / 15

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 ) 4 / 15

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 ‘idempotent subunits are side-effect-free observations’ 4 / 15

Example: logic ISub(Sh( X )) = { S ֌ 1 } = { S ⊆ X | S open } ∈ Frame ‘idempotent subunits are truth values’ 5 / 15

Example: algebra ISub( Mod R ) = � � � � S = S 2 = { x 1 y 1 + · · · x n y n | x i , y i ∈ S } S ⊆ R ideal ‘idempotent subunits are idempotent ideals’ 6 / 15

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 } ‘idempotent subunits are open subsets of base space’ 7 / 15

Example: geometry Hilbert bundle is bundle E ։ X with Hilbert spaces for fibres ISub( Hilb X ) = { S ⊆ X open } ‘idempotent subunits are open subsets of base space’ 8 / 15

Semilattice Proposition: ISub( C ) is a semilattice, ∧ = ⊗ , 1 = id I T I S Caveat: C must be firm, i.e. s ⊗ id T monic, and size issue 9 / 15

Semilattice Proposition: ISub( C ) is a semilattice, ∧ = ⊗ , 1 = id I T I S Caveat: C must be firm, i.e. s ⊗ id T monic, and size issue id ⊥ SemiLat FirmCat ISub 9 / 15

Spatial categories Call C spatial when ISub( C ) is frame SemiLat ⊥ Frame ⊣ ⊣ ISub ISub ? ⊥ SpatCat FirmCat 10 / 15

Spatial categories Call C spatial when ISub( C ) is frame SemiLat ⊥ Frame ⊣ ⊣ ISub ISub ? ⊥ SpatCat FirmCat Idea: � C = [ C op , Set ] is cocomplete 10 / 15

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 / 15

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 / 15

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I id ⊗ s 11 / 15

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I id ⊗ s { s | s supports f } f supp C 2 Pow(ISub( C )) 11 / 15

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I id ⊗ s Monoidal functor: supp( f ) ∧ supp( g ) ≤ supp( f ⊗ g ) { s | s supports f } f supp C 2 Pow(ISub( C )) 11 / 15

Support Say s ∈ ISub( C ) supports f : A → B when A B f ≃ B ⊗ S B ⊗ I id ⊗ s Monoidal functor: supp( f ) ∧ supp( g ) ≤ supp( f ⊗ g ) { s | s supports f } f supp C 2 Pow(ISub( C )) � F F Q ∈ Frame universal with F ( f ) = � { F ( s ) | s ∈ ISub( C ) supports f } 11 / 15

Spatial categories Call F : C op → Set supported when F ( A ) ≃ { f : A → B | supp( f ) ∩ U � = ∅} for some B ∈ C and U ⊆ ISub( C ). ⊥ SemiLat Frame ⊣ ISub ⊣ ISub ⊥ SpatCat FirmCat [ C op , Set ] supp C � = Sh( C , J )! 12 / 15

Complements s Subunit is split when id S I SISub( C ) is a sub-semilattice of ISub( C ) (don’t need firmness) 13 / 15

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 13 / 15

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?) 13 / 15

Linear logic if T : C → C monoidal monad, Kl( T ) is monoidal semilattice morphism { η I ◦ s | s ∈ ISub( C ) , T ( s ) is monic in C } → ISub(Kl( T )) is not injective, nor surjective 14 / 15

Linear logic if T : C → C monoidal monad, Kl( T ) is monoidal semilattice morphism { η I ◦ s | s ∈ ISub( C ) , T ( s ) is monic in C } → ISub(Kl( T )) is not injective, nor surjective model for linear logic: ∗ -autonomous category C with finite products, monoidal comonad !: ( C , ⊗ ) → ( C , × ) (then Kl(!) cartesian closed) if ε epi, then ISub( C , × ) ≃ ISub(Kl(!) , × ) (but hard to compare to ISub( C , ⊗ )) 14 / 15

Further Do you work with maps into a tensor unit? ◮ causality ◮ proof analysis ◮ control flow ◮ data flow ◮ concurrency ◮ graphical calculus 15 / 15

Restriction � � The full subcategory C s of A with id 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 id A ⊗ ε I iso for A ∈ C s )

Restriction � � The full subcategory C s of A with id 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 id A ⊗ ε I iso for A ∈ C s ) Proposition : ISub( C ) ≃ { monocoreflective tensor ideals in C }

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 = { id E ⊗ s | E ∈ C } ( − ) ⊗ S � � s = C [Σ − 1 ] C C ≃ F inverting Σ D