1. Overview: essential surjectivity U We show dd Bicat s -Cat w BrMonCat is essentially surjective on objects: • start with a braided monoidal category B (with strict ⊗ wlog), • construct a dd Bicat s -category Σ B , and ∼ • a braided monoidal equivalence B Σ B . How do we get horizontal and vertical composition from ⊗ ? For tricategories: Put both as ⊗ and get interchange from the braiding γ Issues: 1. We can’t have both compositions strict so they can’t both be ⊗ . 2. We want interchange to be strict so it can’t be γ . 7.
1. Overview: essential surjectivity U We show dd Bicat s -Cat w BrMonCat is essentially surjective on objects: • start with a braided monoidal category B (with strict ⊗ wlog), • construct a dd Bicat s -category Σ B , and ∼ • a braided monoidal equivalence B Σ B . How do we get horizontal and vertical composition from ⊗ ? For tricategories: Put both as ⊗ and get interchange from the braiding γ Issues: 1. We can’t have both compositions strict so they can’t both be ⊗ . 2. We want interchange to be strict so it can’t be γ . Solution: Do “weakification” for the vertical direction. 7.
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. 8.
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). 8.
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). So • F 1 splits into connected components C n “bracketed words of length n ”.
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). So • F 1 splits into connected components C n “bracketed words of length n ”. • Each C n ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”. 8.
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). So • F 1 splits into connected components C n “bracketed words of length n ”. • Each C n ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”. Given a monoidal category M we construct st M “strictification” 8.
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). So • F 1 splits into connected components C n “bracketed words of length n ”. • Each C n ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”. Given a monoidal category M we construct st M “strictification” • objects: words in the objects of M (unbracketed)
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). So • F 1 splits into connected components C n “bracketed words of length n ”. • Each C n ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”. Given a monoidal category M we construct st M “strictification” • objects: words in the objects of M (unbracketed) • morphisms: evaluate words in M then take morphisms from M .
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). So • F 1 splits into connected components C n “bracketed words of length n ”. • Each C n ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”. Given a monoidal category M we construct st M “strictification” • objects: words in the objects of M (unbracketed) • morphisms: evaluate words in M then take morphisms from M . Question: How do we evaluate strict words in a weak monoidal category? 8.
2. Warm-up: strictification of weak monoidal categories Coherence: Every weak monoidal category is monoidal equivalent to a strict one. Follows from: F 1 is monoidal equivalent to the discrete ( N , + , 0). So • F 1 splits into connected components C n “bracketed words of length n ”. • Each C n ≃ 1 so all bracketings are uniquely isomorphic “all diagrams commute”. Given a monoidal category M we construct st M “strictification” • objects: words in the objects of M (unbracketed) • morphisms: evaluate words in M then take morphisms from M . Question: How do we evaluate strict words in a weak monoidal category? Answer: Use cliques. 8.
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . 9.
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . 9.
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . 9.
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . 9. • We could all take different ones by throwing in copies of I .
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . • We could all take different ones by throwing in copies of I . Key: they’re not really different. 9.
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . • We could all take different ones by throwing in copies of I . Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. 9.
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . • We could all take different ones by throwing in copies of I . Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. We just have to agree that we’ve got the same morphism if they only differ by that unique isomorphism: 9.
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . • We could all take different ones by throwing in copies of I . Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. ( f ⊗ g ) ⊗ h ( a ′ ⊗ b ′ ) ⊗ c ( a ⊗ b ) ⊗ c We just have to agree that we’ve got the same morphism if they only differ by that unique isomorphism:
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . • We could all take different ones by throwing in copies of I . Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. ( f ⊗ g ) ⊗ h ( a ′ ⊗ b ′ ) ⊗ c ( a ⊗ b ) ⊗ c We just have to agree that we’ve got the same morphism if they only differ by that unique isomorphism: a ′ ⊗ ( b ′ ⊗ c ′ ) a ⊗ ( b ⊗ c ) f ⊗ ( g ⊗ h )
2. Warm-up: strictification of weak monoidal categories Thought experiment Suppose we’re trying to define morphisms abc y in st M . • I could take morphisms ( a ⊗ b ) ⊗ c y . • You could take morphisms a ⊗ ( b ⊗ c ) y . • We could all take different ones by throwing in copies of I . Key: they’re not really different. All bracketings of abc are uniquely isomorphic via coherence constraints. ( f ⊗ g ) ⊗ h ( a ′ ⊗ b ′ ) ⊗ c ( a ⊗ b ) ⊗ c We just have to agree that we’ve got the same morphism if they only differ ∼ ∼ by that unique isomorphism: a ′ ⊗ ( b ′ ⊗ c ′ ) a ⊗ ( b ⊗ c ) f ⊗ ( g ⊗ h ) 9.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. 10.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · · · · ·
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · Precisely/technically · · A clique in a category C is a functor J C where J ≃ 1 · · 10.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · Precisely/technically · · A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. · · 10.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · Precisely/technically · · A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. · · The unique isomorphisms are called connecting isomorphisms. 10.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · Precisely/technically · · A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. · · The unique isomorphisms are called connecting isomorphisms. A clique map x y is a system of morphisms from each object of x to each object of y making everything commute. 10.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · Precisely/technically · · A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. · · The unique isomorphisms are called connecting isomorphisms. A clique map x y is a system of morphisms from each object of x to each object of y making everything commute. We only have to specify one component to know what all the others are. 10.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · Precisely/technically · · A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. · · The unique isomorphisms are called connecting isomorphisms. · A clique map x y is a system of morphisms from each object of x to each object of y · · making everything commute. We only have to specify one component to know what all the others are. 10.
2. Warm-up: cliques Idea A clique is essentially a collection of objects and unique isomorphisms between them. · Precisely/technically · · A clique in a category C is a functor J C where J ≃ 1 — clique objects are indexed by J but can repeat. · · The unique isomorphisms are called connecting isomorphisms. · A clique map x y is a system of morphisms from each object of x to each object of y · · making everything commute. We only have to specify one component to know what all the others are. 10.
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) 11.
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c )
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) ( a ′ ⊗ b ′ ) ⊗ c a ′ ⊗ ( b ′ ⊗ c ′ ) ∼
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) ( f ⊗ g ) ⊗ h ( a ′ ⊗ b ′ ) ⊗ c a ′ ⊗ ( b ′ ⊗ c ′ ) ∼
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) ( f ⊗ g ) ⊗ h f ⊗ ( g ⊗ h ) ( a ′ ⊗ b ′ ) ⊗ c a ′ ⊗ ( b ′ ⊗ c ′ ) ∼ other components of the same clique map 11.
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) ◦ ( ( f ⊗ g ( f ⊗ g ) ⊗ h f ⊗ ( g ⊗ h ) ) ⊗ h ) ( a ′ ⊗ b ′ ) ⊗ c a ′ ⊗ ( b ′ ⊗ c ′ ) ∼ other components of the same clique map 11.
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) 1 − (( f ⊗ g ) ⊗ h ) ◦ ( f ⊗ g ) ⊗ h f ⊗ ( g ⊗ h ) ( a ′ ⊗ b ′ ) ⊗ c a ′ ⊗ ( b ′ ⊗ c ′ ) ∼ other components of the same clique map 11.
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) We can compose components that don’t ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) look composable via connecting isos 1 − (( f ⊗ g ) ⊗ h ) ◦ ( a ⊗ b ) ⊗ c ( f ⊗ g ) ⊗ h f ⊗ ( g ⊗ h ) ( a ′ ⊗ b ′ ) ⊗ c a ′ ⊗ ( b ′ ⊗ c ′ ) ∼ ( a ′ ⊗ b ′ ) ⊗ c ′ a ′ ⊗ ( b ′ ⊗ c ′ ) other components of the a ′′ ⊗ ( b ′′ ⊗ c ′′ ) same clique map 11.
2. Warm-up: strictification of weak monoidal categories Example There is a clique in M of all bracketings of abc (evaluated) We can compose components that don’t ∼ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) look composable via connecting isos 1 − (( f ⊗ g ) ⊗ h ) ◦ ( a ⊗ b ) ⊗ c a ⊗ ( b ⊗ c ) ( f ⊗ g ) ⊗ h f ⊗ ( g ⊗ h ) ( a ′ ⊗ b ′ ) ⊗ c a ′ ⊗ ( b ′ ⊗ c ′ ) ∼ ( a ′ ⊗ b ′ ) ⊗ c ′ a ′ ⊗ ( b ′ ⊗ c ′ ) other components of the ( a ′′ ⊗ b ′′ ) ⊗ c ′′ a ′′ ⊗ ( b ′′ ⊗ c ′′ ) same clique map 11.
2. Warm-up: dots in boxes 12.
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. 12.
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. Braids come from configurations of points in R 2 , and paths. 12.
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. Braids come from configurations of points in R 2 , and paths. · · · We use configurations of points in the interior of I 2 · 12.
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. Braids come from configurations of points in R 2 , and paths. · · · We use configurations of points in the interior of I 2 · • Vertical composition · · · · · · · · 12.
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. Braids come from configurations of points in R 2 , and paths. · · · We use configurations of points in the interior of I 2 · • Vertical composition · · · · · · · · • Horizontal composition · · · · · · · · 12.
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. Braids come from configurations of points in R 2 , and paths. · · · We use configurations of points in the interior of I 2 · • Vertical composition Problem: Both weak. · · · · · · · · • Horizontal composition · · · · · · · · 12.
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. Braids come from configurations of points in R 2 , and paths. · · · We use configurations of points in the interior of I 2 · • Vertical composition Problem: Both weak. · · Solution: Take “horizontal path” classes · · — paths that do not change any y coordinate · · · · · · • Horizontal composition · · · · · · · · · ·
2. Warm-up: dots in boxes Coherence for braided monoidal categories relates F 1 to the braid category. Braids come from configurations of points in R 2 , and paths. · · · We use configurations of points in the interior of I 2 · • Vertical composition Problem: Both weak. · · Solution: Take “horizontal path” classes · · — paths that do not change any y coordinate · · · · · · • Horizontal composition “strictification in the horizontal direction” · · · · · · · · · · 12.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B 13.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k 13.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k Morphisms (cf strictification): start by evaluating the configuration as a word 13.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k Morphisms (cf strictification): start by evaluating the configuration as a word • similarity: there are many different ways to do so • difference: they are not uniquely isomorphic “not all diagrams commute” 13.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k Morphisms (cf strictification): start by evaluating the configuration as a word • similarity: there are many different ways to do so • difference: they are not uniquely isomorphic “not all diagrams commute” · a a ⊗ b · b 13.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k Morphisms (cf strictification): start by evaluating the configuration as a word • similarity: there are many different ways to do so • difference: they are not uniquely isomorphic “not all diagrams commute” · a a ⊗ b · b a b ? · ·
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k Morphisms (cf strictification): start by evaluating the configuration as a word • similarity: there are many different ways to do so • difference: they are not uniquely isomorphic “not all diagrams commute” · a a ⊗ b · b clockwise a ⊗ b a b ? · · anti-clockwise b ⊗ a 13.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k Morphisms (cf strictification): start by evaluating the configuration as a word • similarity: there are many different ways to do so • difference: they are not uniquely isomorphic “not all diagrams commute” There are many isomorphisms · a a ⊗ b connecting these eg · b clockwise a ⊗ b a b ? · · anti-clockwise b ⊗ a 13.
3. Construction of Σ B Aim: construct a dd Bicat s -category Σ B from a braided monoidal category B a 1 · · · . Objects: horizontal path classes of points in I 2 , labelled by objects of B . . · · a k Morphisms (cf strictification): start by evaluating the configuration as a word • similarity: there are many different ways to do so • difference: they are not uniquely isomorphic “not all diagrams commute” There are many isomorphisms · a a ⊗ b connecting these eg · b clockwise a ⊗ b a b ? · · anti-clockwise b ⊗ a Solution: remember the journey, not just the destination. 13.
3. Construction of Σ B · a · a • The free braided monoidal category embeds vertically: · b · · · · b · c a ⊗ b a ⊗ b ⊗ c 14.
3. Construction of Σ B · a · a • The free braided monoidal category embeds vertically: · b · · · · b · c a ⊗ b a ⊗ b ⊗ c • We “flatten” our configuration to a canonical vertical one and remember what braid we used to do it. · · · · · · · · “flattening braid”
3. Construction of Σ B · a · a • The free braided monoidal category embeds vertically: · b · · · · b · c a ⊗ b a ⊗ b ⊗ c • We “flatten” our configuration to a canonical vertical one and remember what braid we used to do it. · · · · · · α · · · · · · clockwise “flattening braid” a ⊗ b
3. Construction of Σ B · a · a • The free braided monoidal category embeds vertically: · b · · · · b · c a ⊗ b a ⊗ b ⊗ c • We “flatten” our configuration to a canonical vertical one and remember what braid we used to do it. · · · · · · · · α β · · · · · · · · clockwise anti-clockwise “flattening braid” a ⊗ b b ⊗ a
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