Let m n be the full subcategory of spanned by opetopes of dimension between m and n . The category of opetopes Let O be the category whose objects are opetopes and morphisms are source and target embeddings , e.g. . . ⇓ s � → . . . . ⇛ ⇓ ⇓ . . ⇓ ⇓ . . . . . . t . . . . � → ⇓ ⇓ ⇓ ⇛ . . ⇓ . . . . 13
The category of opetopes Let O be the category whose objects are opetopes and morphisms are source and target embeddings , e.g. . . ⇓ s � → . . . . ⇛ ⇓ ⇓ . . ⇓ ⇓ . . . . . . t . . . . � → ⇓ ⇓ ⇓ ⇛ . . ⇓ . . . . Let O m , n be the full subcategory of O spanned by opetopes of dimension between m and n . 13
Likewise, et is the cagetogy of presheaves m n m n over m n , or “truncated opetopic sets”. Example 1. We have since 0 1 and thus, raph , the category of directed 0 1 graphs. 2. Likewise, 1 2 is the category of (non-symmetric) collections. op sh sh sh Opetopic sets Let P sh ( O ) = [ O op , S et ] be the category of opetopic sets . 14
Example 1. We have since 0 1 and thus, raph , the category of directed 0 1 graphs. 2. Likewise, 1 2 is the category of (non-symmetric) collections. sh sh Opetopic sets Let P sh ( O ) = [ O op , S et ] be the category of opetopic sets . Likewise, P sh ( O m , n ) = [ O op m , n , S et ] is the cagetogy of presheaves over O m , n , or “truncated opetopic sets”. 14
2. Likewise, 1 2 is the category of (non-symmetric) collections. sh Opetopic sets Let P sh ( O ) = [ O op , S et ] be the category of opetopic sets . Likewise, P sh ( O m , n ) = [ O op m , n , S et ] is the cagetogy of presheaves over O m , n , or “truncated opetopic sets”. Example 1. We have O 0 , 1 = ( ⧫ ⇉ ◾ ) since ◾ = . . and thus, P sh ( O 0 , 1 ) = G raph , the category of directed graphs. 14
Opetopic sets Let P sh ( O ) = [ O op , S et ] be the category of opetopic sets . Likewise, P sh ( O m , n ) = [ O op m , n , S et ] is the cagetogy of presheaves over O m , n , or “truncated opetopic sets”. Example 1. We have O 0 , 1 = ( ⧫ ⇉ ◾ ) since ◾ = . . and thus, P sh ( O 0 , 1 ) = G raph , the category of directed graphs. 2. Likewise, P sh ( O 1 , 2 ) is the category of (non-symmetric) collections. 14
• Let O O be the boundary of . t • Let O t be the target horn of Opetopic sets Some opetopic sets are of particular interest: . . ⇓ . . . . ⇛ ⇓ ⇓ ⇓ . . . . • For ω ∈ O , let O [ ω ] = O ( − ,ω ) be the representable at ω . 15
t • Let O t be the target horn of Opetopic sets Some opetopic sets are of particular interest: . . ⇓ ⋃ . . . . ⇓ ⇓ ⇓ . . . . • For ω ∈ O , let O [ ω ] = O ( − ,ω ) be the representable at ω . • Let ∂ O [ ω ] = O [ ω ] − { ω } be the boundary of ω . 15
Opetopic sets Some opetopic sets are of particular interest: . ⇓ . . ⇓ ⇓ . . • For ω ∈ O , let O [ ω ] = O ( − ,ω ) be the representable at ω . • Let ∂ O [ ω ] = O [ ω ] − { ω } be the boundary of ω . • Let Λ t [ ω ] = ∂ O [ ω ] − { t ω } be the target horn of ω 15
A morphism t f X amounts to forming a pasting diagram of shape with elements of X . Example . . . . t 3 If 3 , then . Thus, a . . . . t 3 morphism X amounts to the choice of 3 composable arrows of X . Target horns Let ω ∈ O , and X ∈ P sh ( O ) . 16
Example . . . . t 3 If 3 , then . Thus, a . . . . t 3 morphism X amounts to the choice of 3 composable arrows of X . Target horns Let ω ∈ O , and X ∈ P sh ( O ) . A morphism f ∶ Λ t [ ω ] � → X amounts to forming a pasting diagram of shape ω with elements of X . 16
Thus, a t 3 morphism X amounts to the choice of 3 composable arrows of X . Target horns Let ω ∈ O , and X ∈ P sh ( O ) . A morphism f ∶ Λ t [ ω ] � → X amounts to forming a pasting diagram of shape ω with elements of X . Example . . . . , then Λ t [ 3 ] = If ω = 3 = . ⇓ . . . . 16
Target horns Let ω ∈ O , and X ∈ P sh ( O ) . A morphism f ∶ Λ t [ ω ] � → X amounts to forming a pasting diagram of shape ω with elements of X . Example . . . . , then Λ t [ 3 ] = If ω = 3 = . Thus, a ⇓ . . . . morphism Λ t [ 3 ] � → X amounts to the choice of 3 composable arrows of X . 16
In our previous example, . . . . h . . . . Lifting against horn inclusions Lifting f ∶ Λ t [ ω ] � → X through O [ ω ] requires to find a compositor for the pasting diagram of f f Λ t [ ω ] X h ω ¯ f O [ ω ] 17
Lifting against horn inclusions Lifting f ∶ Λ t [ ω ] � → X through O [ ω ] requires to find a compositor for the pasting diagram of f f Λ t [ ω ] X h ω ¯ f O [ ω ] In our previous example, . . . . h ω ∶ ↪ ⇓ . . . . 17
An opetopic set X such that H n 1 X , i.e. t X h O has all compositors of n -dimensional pasting diagrams: every pasting diagram of dimension n has a composite . sh Lifting against horn inclusions Let H n = { h ω ∶ Λ t [ ω ] ↪ O [ ω ] ∣ ω ∈ O n } . 18
Lifting against horn inclusions Let H n = { h ω ∶ Λ t [ ω ] ↪ O [ ω ] ∣ ω ∈ O n } . An opetopic set X ∈ P sh ( O ) such that H n + 1 ⊥ X , i.e. ∀ Λ t [ ω ] X h ω ∃ ! O [ ω ] has all compositors of n -dimensional pasting diagrams: every pasting diagram of dimension n has a composite . 18
Pasting diagram of dimension 1 look like this: . . . . If H 2 X , then we have a composition map paths of X arrows of X which looks like a category! But is associative? (no) Lifting against horn inclusions Example Recall that P sh ( O 0 , 1 ) = G raph . Let X ∈ P sh ( O 0 , 1 ) . 19
If H 2 X , then we have a composition map paths of X arrows of X which looks like a category! But is associative? (no) Lifting against horn inclusions Example Recall that P sh ( O 0 , 1 ) = G raph . Let X ∈ P sh ( O 0 , 1 ) . Pasting diagram of dimension 1 look like this: . . . . 19
But is associative? (no) Lifting against horn inclusions Example Recall that P sh ( O 0 , 1 ) = G raph . Let X ∈ P sh ( O 0 , 1 ) . Pasting diagram of dimension 1 look like this: . . . . If H 2 ⊥ X , then we have a composition map µ ∶ paths of X � → arrows of X which looks like a category! 19
(no) Lifting against horn inclusions Example Recall that P sh ( O 0 , 1 ) = G raph . Let X ∈ P sh ( O 0 , 1 ) . Pasting diagram of dimension 1 look like this: . . . . If H 2 ⊥ X , then we have a composition map µ ∶ paths of X � → arrows of X which looks like a category! But is µ associative? 19
Lifting against horn inclusions Example Recall that P sh ( O 0 , 1 ) = G raph . Let X ∈ P sh ( O 0 , 1 ) . Pasting diagram of dimension 1 look like this: . . . . If H 2 ⊥ X , then we have a composition map µ ∶ paths of X � → arrows of X which looks like a category! But is µ associative? (no) 19
Solution: lift against H n 1 n 2 H n 1 H n 2 . Intuitively, if H n 2 X , then a combination of lifting problems (in dimension n ) can be summarized into a unique one: Lifting against horn inclusions Unfortunately, lifting against H n + 1 does not give an adequate notion of algebra as the composition operation is not associative. 20
Intuitively, if H n 2 X , then a combination of lifting problems (in dimension n ) can be summarized into a unique one: Lifting against horn inclusions Unfortunately, lifting against H n + 1 does not give an adequate notion of algebra as the composition operation is not associative. Solution: lift against H n + 1 , n + 2 = H n + 1 ∪ H n + 2 . 20
Lifting against horn inclusions Unfortunately, lifting against H n + 1 does not give an adequate notion of algebra as the composition operation is not associative. Solution: lift against H n + 1 , n + 2 = H n + 1 ∪ H n + 2 . Intuitively, if H n + 2 ⊥ X , then a combination of lifting problems (in dimension n ) can be summarized into a unique one: . . . ⇓ ⇓ ↪ . . . . . . ⇛ ⇓ ⇓ ⇓ ⇓ ⇓ . . . . . . 20
Then h X ensures that for f g h composable arrows in X we have fg h fgh A similar opetope would enforce f gh fgh . Lifting against horn inclusion Example Let X ∈ P sh ( O ) be an opetopic set such that H 2 , 3 ⊥ X , and consider ⎛ ⎞ . . . . ⎜ ⎟ ω = ⎜ ⎟ ∈ O 3 ⇓ ⇓ ⇛ ⎝ . ⎠ ⇓ . . . 21
A similar opetope would enforce f gh fgh . Lifting against horn inclusion Example Let X ∈ P sh ( O ) be an opetopic set such that H 2 , 3 ⊥ X , and consider ⎛ ⎞ . . . . ⎜ ⎟ ω = ⎜ ⎟ ∈ O 3 ⇓ ⇓ ⇛ ⎝ . ⎠ ⇓ . . . Then h ω ⊥ X ensures that for f , g , h composable arrows in X we have ( fg ) h = fgh . 21
Lifting against horn inclusion Example Let X ∈ P sh ( O ) be an opetopic set such that H 2 , 3 ⊥ X , and consider ⎛ ⎞ . . . . ⎜ ⎟ ω = ⎜ ⎟ ∈ O 3 ⇓ ⇓ ⇛ ⎝ . ⎠ ⇓ . . . Then h ω ⊥ X ensures that for f , g , h composable arrows in X we have ( fg ) h = fgh . A similar opetope would enforce f ( gh ) = fgh . 21
• H n 2 X ensures that it is suitably associative. The last step required to define opetopic algebra is to trivialize X in dimension n and n 2. Opetopic algebras (almost) So to summarize: • H n + 1 ⊥ X gives a composition operation for n -dimensional cells of X ; 22
The last step required to define opetopic algebra is to trivialize X in dimension n and n 2. Opetopic algebras (almost) So to summarize: • H n + 1 ⊥ X gives a composition operation for n -dimensional cells of X ; • H n + 2 ⊥ X ensures that it is suitably associative. 22
Opetopic algebras (almost) So to summarize: • H n + 1 ⊥ X gives a composition operation for n -dimensional cells of X ; • H n + 2 ⊥ X ensures that it is suitably associative. The last step required to define opetopic algebra is to trivialize X in dimension < n and > n + 2. 22
Solution: require O n X , where O n O n • We want X to be “trivial” in dimension n 2. Solution: require B n 2 X , where B n 2 O O n 2 Lemma H n 1 n 2 B n 2 X H n 1 X Trivialization • We want X to be “trivial” in dimension < n . 23
• We want X to be “trivial” in dimension n 2. Solution: require B n 2 X , where B n 2 O O n 2 Lemma H n 1 n 2 B n 2 X H n 1 X Trivialization • We want X to be “trivial” in dimension < n . Solution: require O < n ⊥ X , where O < n = { ∅ ↪ O [ ψ ] ∣ dim ψ < n } . 23
Solution: require B n 2 X , where B n 2 O O n 2 Lemma H n 1 n 2 B n 2 X H n 1 X Trivialization • We want X to be “trivial” in dimension < n . Solution: require O < n ⊥ X , where O < n = { ∅ ↪ O [ ψ ] ∣ dim ψ < n } . • We want X to be “trivial” in dimension > n + 2. 23
Lemma H n 1 n 2 B n 2 X H n 1 X Trivialization • We want X to be “trivial” in dimension < n . Solution: require O < n ⊥ X , where O < n = { ∅ ↪ O [ ψ ] ∣ dim ψ < n } . • We want X to be “trivial” in dimension > n + 2. Solution: require B > n + 2 ⊥ X , where B > n + 2 = { ∂ O [ ψ ] ↪ O [ ψ ] ∣ dim ψ > n + 2 } 23
Trivialization • We want X to be “trivial” in dimension < n . Solution: require O < n ⊥ X , where O < n = { ∅ ↪ O [ ψ ] ∣ dim ψ < n } . • We want X to be “trivial” in dimension > n + 2. Solution: require B > n + 2 ⊥ X , where B > n + 2 = { ∂ O [ ψ ] ↪ O [ ψ ] ∣ dim ψ > n + 2 } Lemma H n + 1 , n + 2 ∪ B > n + 2 ⊥ X H ≥ n + 1 ⊥ X ⇐ ⇒ 23
Examples • Monoids are exactly 0 1 -opetopic algebras. • Planar uncolored operads are exactly 0 2 -opetopic algebras. • Loday’s combinads (over the combinatorial pattern of planar trees) are exactly 0 3 -opetopic algebras. Opetopic algebras Definition A ( 0 , n ) -opetopic algebra is an opetopic set X such that O < n ∪ H ≥ n + 1 ⊥ X . 24
• Planar uncolored operads are exactly 0 2 -opetopic algebras. • Loday’s combinads (over the combinatorial pattern of planar trees) are exactly 0 3 -opetopic algebras. Opetopic algebras Definition A ( 0 , n ) -opetopic algebra is an opetopic set X such that O < n ∪ H ≥ n + 1 ⊥ X . Examples • Monoids are exactly ( 0 , 1 ) -opetopic algebras. 24
• Loday’s combinads (over the combinatorial pattern of planar trees) are exactly 0 3 -opetopic algebras. Opetopic algebras Definition A ( 0 , n ) -opetopic algebra is an opetopic set X such that O < n ∪ H ≥ n + 1 ⊥ X . Examples • Monoids are exactly ( 0 , 1 ) -opetopic algebras. • Planar uncolored operads are exactly ( 0 , 2 ) -opetopic algebras. 24
Opetopic algebras Definition A ( 0 , n ) -opetopic algebra is an opetopic set X such that O < n ∪ H ≥ n + 1 ⊥ X . Examples • Monoids are exactly ( 0 , 1 ) -opetopic algebras. • Planar uncolored operads are exactly ( 0 , 2 ) -opetopic algebras. • Loday’s combinads (over the combinatorial pattern PT of planar trees) are exactly ( 0 , 3 ) -opetopic algebras. 24
Solution: Don’t trivialize low dimensions as much: Definition A k-colored n-opetopic algebra (or simply k n -opetopic algebra) is an opetopic set X such that O n k H n 1 X A k n Examples • Categories (colored monoids) are exactly 1 1 -opetopic algebras. • Planar colored operads are exactly 1 2 -opetopic algebras. Opetopic algebras What if we want some colors in our algebras? 25
Examples • Categories (colored monoids) are exactly 1 1 -opetopic algebras. • Planar colored operads are exactly 1 2 -opetopic algebras. Opetopic algebras What if we want some colors in our algebras? Solution: Don’t trivialize low dimensions as much: Definition A k-colored n-opetopic algebra (or simply ( k , n ) -opetopic algebra) is an opetopic set X such that O < n − k ∪ H ≥ n + 1 ⊥ X . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� A k , n 25
• Planar colored operads are exactly 1 2 -opetopic algebras. Opetopic algebras What if we want some colors in our algebras? Solution: Don’t trivialize low dimensions as much: Definition A k-colored n-opetopic algebra (or simply ( k , n ) -opetopic algebra) is an opetopic set X such that O < n − k ∪ H ≥ n + 1 ⊥ X . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� A k , n Examples • Categories (colored monoids) are exactly ( 1 , 1 ) -opetopic algebras. 25
Opetopic algebras What if we want some colors in our algebras? Solution: Don’t trivialize low dimensions as much: Definition A k-colored n-opetopic algebra (or simply ( k , n ) -opetopic algebra) is an opetopic set X such that O < n − k ∪ H ≥ n + 1 ⊥ X . �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� A k , n Examples • Categories (colored monoids) are exactly ( 1 , 1 ) -opetopic algebras. • Planar colored operads are exactly ( 1 , 2 ) -opetopic algebras. 25
Opetopic algebras: monadic approach
We now describe the “free k n -algebra”-monad, which constructs all those pasting diagrams. Intuition: back to pasting diagrams Recall that if X is a ( k , n ) -algebra, then there is a composition operation µ ∶ { n -dim. pasting diags. of X } � → X 26
Intuition: back to pasting diagrams Recall that if X is a ( k , n ) -algebra, then there is a composition operation µ ∶ { n -dim. pasting diags. of X } � → X We now describe the “free ( k , n ) -algebra”-monad, which constructs all those pasting diagrams. 26
n does not act on colors , we have n Y Y for • Since n k n 1 . • Let n 1 be A pasting diagram as on the left ( t ) needs to be evaluated to a cell as on the right (t ). Thus for n , n Y t Y n k n n 1 t sh The Z n monad Discarding irrelevant dimensions, we want a monad Z n ∶ P sh ( O n − k , n ) � → P sh ( O n − k , n ) that “constructs pasting diagrams”. 27
• Let n 1 be A pasting diagram as on the left ( t ) needs to be evaluated to a cell as on the right (t ). Thus for n , n Y t Y n k n n 1 t sh The Z n monad Discarding irrelevant dimensions, we want a monad Z n ∶ P sh ( O n − k , n ) � → P sh ( O n − k , n ) that “constructs pasting diagrams”. • Since Z n does not act on colors , we have Z n Y φ = Y φ for φ ∈ O n − k , n − 1 . 27
A pasting diagram as on the left ( t ) needs to be evaluated to a cell as on the right (t ). Thus for n , n Y t Y n k n n 1 t sh The Z n monad Discarding irrelevant dimensions, we want a monad Z n ∶ P sh ( O n − k , n ) � → P sh ( O n − k , n ) that “constructs pasting diagrams”. • Since Z n does not act on colors , we have Z n Y φ = Y φ for φ ∈ O n − k , n − 1 . . . ⇓ • Let ω ∈ O n + 1 be . . . . ⇛ ⇓ ⇓ ⇓ . . . . 27
Thus for n , n Y t Y n k n n 1 t sh The Z n monad Discarding irrelevant dimensions, we want a monad Z n ∶ P sh ( O n − k , n ) � → P sh ( O n − k , n ) that “constructs pasting diagrams”. • Since Z n does not act on colors , we have Z n Y φ = Y φ for φ ∈ O n − k , n − 1 . . . ⇓ • Let ω ∈ O n + 1 be A pasting . . . . ⇛ ⇓ ⇓ ⇓ . . . . diagram as on the left ( Λ t [ ω ] ) needs to be evaluated to a cell as on the right (t ω ). 27
The Z n monad Discarding irrelevant dimensions, we want a monad Z n ∶ P sh ( O n − k , n ) � → P sh ( O n − k , n ) that “constructs pasting diagrams”. • Since Z n does not act on colors , we have Z n Y φ = Y φ for φ ∈ O n − k , n − 1 . . . ⇓ • Let ω ∈ O n + 1 be A pasting . . . . ⇛ ⇓ ⇓ ⇓ . . . . diagram as on the left ( Λ t [ ω ] ) needs to be evaluated to a cell as on the right (t ω ). Thus for ψ ∈ O n , Z n Y ψ = P sh ( O n − k , n )( Λ t [ ω ] , Y ) . ∑ ω ∈ O n + 1 t ω = ψ 27
Proof (sketch) n Y : a single cell of Y is already a pasting • Unit Y diagram. n n Y n Y : a pasting diagram of pasting • Multiplication diagrams is a pasting diagram. n the Eilenberg–Moore category of lg k We write n n k n . n k n sh sh The Z n monad Theorem The endofunctor Z n is canonically a parametric right adjoint monad. 28
n n Y n Y : a pasting diagram of pasting • Multiplication diagrams is a pasting diagram. n the Eilenberg–Moore category of lg k We write n n k n . n k n sh sh The Z n monad Theorem The endofunctor Z n is canonically a parametric right adjoint monad. Proof (sketch) → Z n Y : a single cell of Y is already a pasting • Unit Y � diagram. 28
n the Eilenberg–Moore category of lg k We write n n k n . n k n sh sh The Z n monad Theorem The endofunctor Z n is canonically a parametric right adjoint monad. Proof (sketch) → Z n Y : a single cell of Y is already a pasting • Unit Y � diagram. • Multiplication Z n Z n Y � → Z n Y : a pasting diagram of pasting diagrams is a pasting diagram. 28
The Z n monad Theorem The endofunctor Z n is canonically a parametric right adjoint monad. Proof (sketch) → Z n Y : a single cell of Y is already a pasting • Unit Y � diagram. • Multiplication Z n Z n Y � → Z n Y : a pasting diagram of pasting diagrams is a pasting diagram. We write A lg k ( Z n ) the Eilenberg–Moore category of Z n ∶ P sh ( O n − k , n ) � → P sh ( O n − k , n ) . 28
Theorem There is an adjunction lg k n h k n N k n n as the localization A 1 lg k that exhibits . In other k n sh n algebras are the same! words, k n -algebras and sh Opetopic algebras: monadic definition Recall that ( k , n ) -opetopic algebras are opetopic sets X ∈ P sh ( O ) such that A k , n ⊥ X . 29
Opetopic algebras: monadic definition Recall that ( k , n ) -opetopic algebras are opetopic sets X ∈ P sh ( O ) such that A k , n ⊥ X . Theorem There is an adjunction � A lg k ( Z n ) ∶ N k , n h k , n ∶ P sh ( O ) � → ← that exhibits A lg k ( Z n ) as the localization A − 1 k , n P sh ( O ) . In other words, ( k , n ) -algebras and Z n algebras are the same! 29
• If k n 1 1 , then raph , and n k n 1 raph is the free category monad. • If k n 0 2 , then et , and n k n 2 is the free uncolored planar operad monad. • If k n 1 2 , then oll is the category of n k n 2 (non symmetric) collections, and oll is the free colored planar operad monad. So we have an infinite hierarchy of “higher arity algebras”! (no) sh sh et oll sh raph et Opetopic algebras: monadic definition Examples • If ( k , n ) = ( 0 , 1 ) , then P sh ( O n − k , n ) = S et , and Z 1 ∶ S et � → S et is the free monoid monad. 30
• If k n 0 2 , then et , and n k n 2 is the free uncolored planar operad monad. • If k n 1 2 , then oll is the category of n k n 2 (non symmetric) collections, and oll is the free colored planar operad monad. So we have an infinite hierarchy of “higher arity algebras”! (no) et sh et oll sh Opetopic algebras: monadic definition Examples • If ( k , n ) = ( 0 , 1 ) , then P sh ( O n − k , n ) = S et , and Z 1 ∶ S et � → S et is the free monoid monad. • If ( k , n ) = ( 1 , 1 ) , then P sh ( O n − k , n ) = G raph , and Z 1 ∶ G raph � → G raph is the free category monad. 30
• If k n 1 2 , then oll is the category of n k n 2 (non symmetric) collections, and oll is the free colored planar operad monad. So we have an infinite hierarchy of “higher arity algebras”! (no) oll sh Opetopic algebras: monadic definition Examples • If ( k , n ) = ( 0 , 1 ) , then P sh ( O n − k , n ) = S et , and Z 1 ∶ S et � → S et is the free monoid monad. • If ( k , n ) = ( 1 , 1 ) , then P sh ( O n − k , n ) = G raph , and Z 1 ∶ G raph � → G raph is the free category monad. • If ( k , n ) = ( 0 , 2 ) , then P sh ( O n − k , n ) = S et N , and → S et N is the free uncolored planar operad Z 2 ∶ S et N � monad. 30
So we have an infinite hierarchy of “higher arity algebras”! (no) Opetopic algebras: monadic definition Examples • If ( k , n ) = ( 0 , 1 ) , then P sh ( O n − k , n ) = S et , and Z 1 ∶ S et � → S et is the free monoid monad. • If ( k , n ) = ( 1 , 1 ) , then P sh ( O n − k , n ) = G raph , and Z 1 ∶ G raph � → G raph is the free category monad. • If ( k , n ) = ( 0 , 2 ) , then P sh ( O n − k , n ) = S et N , and → S et N is the free uncolored planar operad Z 2 ∶ S et N � monad. • If ( k , n ) = ( 1 , 2 ) , then P sh ( O n − k , n ) = C oll is the category of (non symmetric) collections, and Z 2 ∶ C oll � → C oll is the free colored planar operad monad. 30
(no) Opetopic algebras: monadic definition Examples • If ( k , n ) = ( 0 , 1 ) , then P sh ( O n − k , n ) = S et , and Z 1 ∶ S et � → S et is the free monoid monad. • If ( k , n ) = ( 1 , 1 ) , then P sh ( O n − k , n ) = G raph , and Z 1 ∶ G raph � → G raph is the free category monad. • If ( k , n ) = ( 0 , 2 ) , then P sh ( O n − k , n ) = S et N , and → S et N is the free uncolored planar operad Z 2 ∶ S et N � monad. • If ( k , n ) = ( 1 , 2 ) , then P sh ( O n − k , n ) = C oll is the category of (non symmetric) collections, and Z 2 ∶ C oll � → C oll is the free colored planar operad monad. So we have an infinite hierarchy of “higher arity algebras”! 30
Opetopic algebras: monadic definition Examples • If ( k , n ) = ( 0 , 1 ) , then P sh ( O n − k , n ) = S et , and Z 1 ∶ S et � → S et is the free monoid monad. • If ( k , n ) = ( 1 , 1 ) , then P sh ( O n − k , n ) = G raph , and Z 1 ∶ G raph � → G raph is the free category monad. • If ( k , n ) = ( 0 , 2 ) , then P sh ( O n − k , n ) = S et N , and → S et N is the free uncolored planar operad Z 2 ∶ S et N � monad. • If ( k , n ) = ( 1 , 2 ) , then P sh ( O n − k , n ) = C oll is the category of (non symmetric) collections, and Z 2 ∶ C oll � → C oll is the free colored planar operad monad. So we have an infinite hierarchy of “higher arity algebras”! (no) 30
The algebraic trompe-l’œil
Recommend
More recommend
