The idea in the 3-dimensional case The 2-dimensional case The tricategory of formal composites and its strictification Peter Guthmann University of Leicester September 11, 2019 Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case 1 The idea in the 3-dimensional case 2 The 2-dimensional case Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case [ˆ [ˆ β ] β ] ∼ [ˆ α ] [ˆ α ] [ˆ α ] [ˆ α ] [ˆ [ˆ Φ] Ψ] [ˆ α ] Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case 1 [ f ][ g ] [ˆ Φ − 1 ] 1 [ˆ α ] [ˆ α ] 11 1 [ˆ β ] 1 [ f ][ g ] [ f ][ g ][ f ][ g ] 1 ∼ 11 [ˆ α ] [ˆ α ] 1 [ˆ Ψ] [ f ][ g ] 1 Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Theorem (Gordon-Power-Street) Every tricategory T is triequivalent to a Gray category Gr T . But the triequivalence T → Gr T is not strict. Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Theorem (Gordon-Power-Street) Every tricategory T is triequivalent to a Gray category Gr T . But the triequivalence T → Gr T is not strict. Theorem (G.) There exists a span of strict triequivalences � T [ − ] ev T st . T Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Definition (bicategory) Collection of objects Ob( B ) , local hom-categories B ( a , b ) for all objects a , b ∈ B , identity functors I a : 1 → B ( a , a ) , composition functors ∗ a , b , c : B ( b , c ) × B ( a , b ) → B ( a , c ) , and natural transformations a , l , r corresponding to the axioms of a category. Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case ∗× 1 B ( c , d ) × B ( b , c ) × B ( a , b ) B ( b , d ) × B ( a , b ) 1 ×∗ a ∗ B ( c , d ) × B ( a , c ) B ( a , d ) , ∗ B ( b , b ) × B ( a , b ) B ( a , b ) × B ( a , a ) I b × 1 1 × I a ∗ ∗ r l B ( a , b ) B ( a , b ) B ( a , b ) B ( a , b ) . 1 1 Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case The axioms of bicategory are chosen such that a coherence law holds. Proposition (Coherence law) Parallel coherence-morphisms in a free bicategory are equal. (Free on a Cat − graph . ) Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Definition A 2-category is called strict if a , l , r are identity natural transformations In a strict 2-category we can denote 2-cells as follows. ... f 2 f n f 1 f 2 • . . . • • f n f 1 α • α • g m g 1 • • . . . g 2 • ... g m g 1 g 2 Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Proposition (Power) Pasting diagrams are well defined for 2-categories. (And thus string diagrams are.) Example: interchange law • • • Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Folklore ’theorem’: Pasting diagrams / String diagrams work also in bicategories. Fix a source fix a target insert coherence cells as needed and the resulting 2-cell will be well defined. I.e. independent of the choice of inserted constraint cells. How can this made precise? Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Proposition There exists a bicategory � B and a strict 2-category B st together with strict biequivalences as in the following diagram. � B [ − ] ev B st . B How does � B look like? Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Definition Let B be a bicategory. Then the following defines a bicategory � B together with a strict biequivalence ev : � B → B : Ob( � B ) = Ob( B ) and ev acts on objects as an identity. The 1-morphisms of � B are formal composites of 1-morphisms in B . Thus a generic 2-morphisms looks like: � � f ˆ ( g ˆ ∗ h ) ˆ ∗ ( k ˆ ∗ l ) ∗ . The action of ev on 1-morphisms is given by evaluation. For example � � � � f ˆ ∗ (( g ˆ ∗ h ) ˆ ∗ ( k ˆ ev ∗ l )) = f ∗ ( g ∗ h ) ∗ ( k ∗ l ) . Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Definition B are triples ( α, ˆ g ) : ˆ The 2-morphisms of � f , ˆ f → ˆ g where α : ev ˆ f → ev ˆ g is a 2-morphism in B . ev acts on 2-morphisms via ev( α, ˆ f , ˆ g ) = α. The constraint-cells of � B are given by � � h ) , (ˆ ∗ ˆ h , ˆ ∗ ˆ f ˆ g ) ˆ f ˆ g ˆ ˆ a ˆ h = a ev(ˆ ∗ ˆ ∗ (ˆ h ) g ˆ g ) ev(ˆ f ˆ f ) ev(ˆ ˆ ∗ ˆ f , ˆ f ) , ˆ f , ˆ f ) , ˆ ∗ ˆ f = ( l ev(ˆ 1 t ˆ f ˆ f ) and ˆ f = ( r ev(ˆ f ˆ 1 s ˆ f ) l ˆ r ˆ Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Parallel coherence morphisms in � B are equal. Thus coherence can be quotient out of � B which leads to a 2-category B st . Taking equivalence classes gives the desired strict biequivalence [ − ] : B → B st . Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case How can the span of strict biequivalences � B [ − ] ev B st . B be used to reduce calculations in bicategories to calculations in 2-categories. Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Example: Adjunction in bicategory B Data: 1-morphism f : a → b and g : b → a 2-morphism η : 1 a → g ∗ f and ǫ : f ∗ g → 1 b . Axioms: r − 1 → f ∗ ( g ∗ f ) a − 1 1 ∗ η ǫ ∗ 1 l 1 f → f ∗ 1 → ( f ∗ g ) ∗ f → 1 ∗ f → f = f → f − − − − − − − − − − (1) l − 1 η ∗ 1 → g ∗ ( f ∗ g ) 1 ∗ ǫ 1 a → g ∗ 1 r → 1 ∗ g → ( g ∗ f ) ∗ g → g = g g → g . − − − − − − − − − (2) Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Lift of the adjunction along ev : � B → B : Data: 1-morphism f : a → b and g : b → a η := ( η, 1 a , g ˆ ǫ := ( ǫ, f ˆ 2-morphism ˆ ∗ f ) and ˆ ∗ g , 1 b ) Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Lift of the adjunction along ev : � B → B : Data: 1-morphism f : a → b and g : b → a η := ( η, 1 a , g ˆ ǫ := ( ǫ, f ˆ 2-morphism ˆ ∗ f ) and ˆ ∗ g , 1 b ) Axioms: ˆ ∗ ˆ ˆ ˆ r − 1 1 ˆ ∗ ˆ a − 1 ˆ η ∗ f ) ˆ ǫ ˆ ˆ 1 l 1 f → f ˆ ∗ 1 → f ˆ ∗ ( g ˆ → ( f ˆ ∗ g ) ˆ ∗ f → 1 ˆ ∗ f → f = f → f − − − − − − − − − − (3) ˆ ∗ ˆ l − 1 ˆ ˆ ˆ η ˆ 1 ˆ 1 ˆ ∗ ˆ ∗ 1 ˆ a ǫ r 1 → 1 ˆ → ( g ˆ ∗ f ) ˆ → g ˆ ∗ ( f ˆ ∗ g ) → g ˆ → g = g g ∗ g ∗ g → g . − − − − − − − − − (4) Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case B → B st : The adjunction in � B under [ − ] : � [ˆ η ] [ˆ η ] [ g ] = [ f ] = and (5) [ f ] [ g ] [ f ] [ f ] [ g ] [ g ] [ˆ ǫ ] [ˆ ǫ ] Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case Lemma Let ( f , g , η, ǫ ) be an equivalence in a bicategory B . Then the equivalence ( f , g , η, ǫ ) satisfies both equations 1 and 2 if it satisfies one of it. Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
The idea in the 3-dimensional case The 2-dimensional case [ˆ η ] [ˆ η ] [ˆ η ] ǫ ) − 1 ] [(ˆ ǫ ) − 1 ] [(ˆ = = = [ˆ ǫ ] [ˆ ǫ ] [ˆ ǫ ] [ˆ ǫ ] [ˆ ǫ ] = = = . Peter Guthmann University of Leicester The tricategory of formal composites and its strictification
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