Two 2-traces Simon Willerton University of Sheffield f Tr ց ( f ) := θ V Tr � ( f ) := V f
Traces What is a trace? Tr ( f ◦ g ) = Tr ( g ◦ f ) Tr ( f ) = Tr ( a ◦ f ◦ a − 1 )
Traces in a monoidal category In ( C , ⊗ , 1 ) , an object V ∗ is left-dual to V if there exist morphisms V ∗ V V ∗ V ev − V ∗ ⊗ V coev V ⊗ V ∗ ← ← − − 1 1 such that V ∗ V ∗ V = = V If V is also left dual to V ∗ then V and V ∗ are bidual. f If V has a bidual and V ← − V define Tr ( f ) := ∈ Hom ( 1 , 1 ) . f V In ( Vect , ⊗ , C ) this gives the usual trace on finite dimensional vector spaces.
Transposes (or adjoints or duals) f If V and W have biduals then V ← − W has a transpose (or is cyclic) if W ∗ f W ∗ f ∗ V ∗ f V ∗ = =: W ∗ V ∗ Theorem (Trace property) g f If V ← − W and W ← − V with f having a transpose then f ∗ Tr ( f ◦ g ) = = f g g = = Tr ( g ◦ f ) g f
� � � � � � � Examples of monoidal bicategories objects 1-morphisms composition 2-morphisms T × Y S � T ′ � T � � � � � � � � � T Span Sets � � � × � T � S � � � � � � � � � � � � � � � � � � Y X � � Y X Z Y X Hom B , A ( B M A , B M ′ Bim Algebras/ C B M A C N B ⊗ B B M A A ) ⊗ C C op ⊗ D → V V -Mod V -cats ⊗ D V -nat trans ⊗ 2-Tang pts in plane cobordisms F E • Ext • Y × X ( E • , F • ) ↓ × Var C -manifolds convolution Y × X ⊗ L ⊗ C DBim Diff algs/ C → B M i A → B M i − 1 Ext • B × A op ( B M • A , B N • → A ) B A
Biduals in a monoidal bicategory In C , an object V ∗ is left-dual to V if there exist 1-morphisms V ∗ V V ∗ V ev − V ∗ ⊗ V coev V ⊗ V ∗ 1 ← ← − − 1 and 2-isomorphisms ∼ ∼ V ∗ V ∗ V ⇒ ⇒ V such that the Swallowtail Relations hold, e.g., ⇒ ⇒ ⇒ Id = ⇒ . If V is also left dual to V ∗ then V and V ∗ are bidual.
Transposes in monoidal bicategories f A 1-morphism V ← − W has a transpose (or is cyclic) if there is a 1-morphism f ∗ W ∗ − V ∗ : ← W ∗ f ∗ V ∗ together with isomorphisms ∼ W ∗ f ∗ ∼ W ∗ f V ∗ f V ∗ ⇒ ⇐ W ∗ V ∗ satisfying some conditions. This gives for example ∼ f ∗ ⇒ f
� � � Examples of duals in monoidal bicategories object bidual evaluation morphism transpose X � T T ∆ Span X X � � � � � � � � � � � � � � � X × X Y X X Y ⋆ A op Bim A C A A ⊗ A op B M A A op M B op C op V -Mod C C op ⊗ C ⊗ ⋆ Hom C op ⊗ D → V ( D op ) op ⊗ C op → V − − − → V 2-Tang E • E • O ∆ Var X X ↓ ↓ ↓ ⋆ × X × X Y × X X × Y A • A • op C A • B • M • A • op M • DBim A • ⊗ A • op A • B • op
The round trace f If V has a bidual and V ← − V define the round trace: Tr � ( f ) := ∈ 1 - Hom ( 1 , 1 ) . V f Theorem (Trace property) f g If V ← − W and W ← − V with f having a transpose then Tr � ( f ◦ g ) ∼ = Tr � ( g ◦ f ) . ∼ Tr � ( f ◦ g ) = f ∗ ⇒ f g g ∼ = Tr � ( g ◦ f ) ⇒ g f
The diagonal trace This can be defined in a bicategory without monoidal structure. f If V is an object of a bicategory and V ← − V define the diagonal trace: f Tr ց ( f ) := 2 - Hom ( Id V , f ) = θ V Theorem (Trace property) a ′ η a − W with a 2 -morphism a ◦ a ′ If W ← − V and V ← ⇐ Id W then you get a (functorial) morphism between sets (or V -objects): η ∗ Tr ց ( f ) → Tr ց ( a ◦ f ◦ a ′ ) − a ′ f a f �→ θ θ V W a In particular if W ← − V is an equivalence then Tr ց ( f ) ∼ = Tr ց ( a ◦ f ◦ a − 1 ) .
� Examples of traces in monoidal bicategories Tr � ( f ) Tr ց ( f ) object endo, f “choice of loop at T Span X � “loops in T ” � � � � each x ∈ X ” X X M / { ma − am } { m ∈ M | am = ma } Bim A A M A coinvariants invariants Z c Z V -Mod C C op ⊗ C F F ( c , c ) F ( c , c ) − → V c 8 9 < = 2-Tang : ; E • HH • ( X , E • ) HH • ( X , E • ) Var X ↓ X × X A • A • M • HH • ( A • , M • ) HH • ( A • , M • ) DBim A •
Dimension The dimension of an object can be defined to be the trace of the identity. Dim � ( V ) := Tr � ( Id V ) = ∈ 1 - Hom ( 1 , 1 ) V � � Dim ց ( V ) := Tr ց ( Id V ) = 2 - Hom ( Id V , Id V ) = θ V ◮ Dim ց ( V ) is a commutative monoid ◮ Dim ց ( V ) acts on Dim � ( V ) Dim ց ( V ) → 2 - Hom Dim � ( V ) , Dim � ( V ) � � �→ θ θ V
Examples of dimensions in monoidal bicategories Dim � ( V ) Dim ց ( V ) object, V Span X X { ⋆ } Bim A A / [ A , A ] Z(A) Z c V -Mod C C ( c , c ) V -NAT ( Id C , Id C ) 2-Tang HH • ( X ) Var X HH • ( X ) A • HH • ( A • ) HH • ( A • ) DBim
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