Two monoidal structures and copoweringDefinition of enriched caterories or internal homs without using...
Two monoidal structures and copowering
Definition of enriched caterories or internal homs without using monoidal categories.Unitalization internal to monoidal categoriesCorrespondence between operads and monads requires tensor distribute over coproduct?Making additive envelopes of monoidal categories monoidalEnriching categories and equivalencesSeeking more information regarding the “rigoidal category” of $mathbb{N}$-graded setsIs there a monoidal category that coclassifies enriched category structures for a given set?Biased vs unbiased lax monoidal categoriesDefinitions of enriched monoidal categoryEnrichment of lax monoidal functors between closed monoidal categories
$begingroup$
Let $(mathbf{M},otimes,1)$ be a closed monoidal category and $(mathbf{C},oplus,0)$ an $mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $odot:mathbf{M}timesmathbf{C}to mathbf{C}$. Is there a canonical morphism
$$(Aodot X)oplus (Bodot Y)to (Aotimes B)odot (Xoplus Y)$$
The question came to my mind because in order to spell out the axioms (in one of the definitions) for an algebra over an operad $mathcal{O}$ in the above setting, we need for the associativity axiom a morphism
$$mathcal{O}(r)odot left(bigoplus_i (mathcal{O}(k_i)odot X^{oplus k_i})right)toleft(mathcal{O}(r)otimesbigotimes_{i}mathcal{O}(k_i)right)odot left(bigoplus_iX^{oplus k_i}right)$$
Or the other direction. If $mathbf{M}$ is considered to be enriched over itself, everything is fine because then $otimes=odot=oplus$, but in general?
ct.category-theory monoidal-categories operads enriched-category-theory
asked 1 hour ago
FKranholdFKranhold
1996
$endgroup$
add a comment |
$begingroup$
Let $(mathbf{M},otimes,1)$ be a closed monoidal category and $(mathbf{C},oplus,0)$ an $mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $odot:mathbf{M}timesmathbf{C}to mathbf{C}$. Is there a canonical morphism
$$(Aodot X)oplus (Bodot Y)to (Aotimes B)odot (Xoplus Y)$$
The question came to my mind because in order to spell out the axioms (in one of the definitions) for an algebra over an operad $mathcal{O}$ in the above setting, we need for the associativity axiom a morphism
$$mathcal{O}(r)odot left(bigoplus_i (mathcal{O}(k_i)odot X^{oplus k_i})right)toleft(mathcal{O}(r)otimesbigotimes_{i}mathcal{O}(k_i)right)odot left(bigoplus_iX^{oplus k_i}right)$$
Or the other direction. If $mathbf{M}$ is considered to be enriched over itself, everything is fine because then $otimes=odot=oplus$, but in general?
ct.category-theory monoidal-categories operads enriched-category-theory
asked 1 hour ago
FKranholdFKranhold
1996
$endgroup$
1
$begingroup$
Okay, it seems to be equivalent to the formulation: “The copowering is a monoidal functor with respect to the component-wise monoidal structure.”
$endgroup$
– FKranhold
1 hour ago
add a comment |
$begingroup$
Let $(mathbf{M},otimes,1)$ be a closed monoidal category and $(mathbf{C},oplus,0)$ an $mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $odot:mathbf{M}timesmathbf{C}to mathbf{C}$. Is there a canonical morphism
$$(Aodot X)oplus (Bodot Y)to (Aotimes B)odot (Xoplus Y)$$
The question came to my mind because in order to spell out the axioms (in one of the definitions) for an algebra over an operad $mathcal{O}$ in the above setting, we need for the associativity axiom a morphism
$$mathcal{O}(r)odot left(bigoplus_i (mathcal{O}(k_i)odot X^{oplus k_i})right)toleft(mathcal{O}(r)otimesbigotimes_{i}mathcal{O}(k_i)right)odot left(bigoplus_iX^{oplus k_i}right)$$
Or the other direction. If $mathbf{M}$ is considered to be enriched over itself, everything is fine because then $otimes=odot=oplus$, but in general?
ct.category-theory monoidal-categories operads enriched-category-theory
asked 1 hour ago
FKranholdFKranhold
1996
$endgroup$
Let $(mathbf{M},otimes,1)$ be a closed monoidal category and $(mathbf{C},oplus,0)$ an $mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $odot:mathbf{M}timesmathbf{C}to mathbf{C}$. Is there a canonical morphism
$$(Aodot X)oplus (Bodot Y)to (Aotimes B)odot (Xoplus Y)$$
The question came to my mind because in order to spell out the axioms (in one of the definitions) for an algebra over an operad $mathcal{O}$ in the above setting, we need for the associativity axiom a morphism
$$mathcal{O}(r)odot left(bigoplus_i (mathcal{O}(k_i)odot X^{oplus k_i})right)toleft(mathcal{O}(r)otimesbigotimes_{i}mathcal{O}(k_i)right)odot left(bigoplus_iX^{oplus k_i}right)$$
Or the other direction. If $mathbf{M}$ is considered to be enriched over itself, everything is fine because then $otimes=odot=oplus$, but in general?
ct.category-theory monoidal-categories operads enriched-category-theory
ct.category-theory monoidal-categories operads enriched-category-theory
asked 1 hour ago
FKranholdFKranhold
1996
asked 1 hour ago
FKranholdFKranhold
1996
asked 1 hour ago
FKranholdFKranhold
1996
asked 1 hour ago
FKranholdFKranhold
1996
asked 1 hour ago
FKranholdFKranhold
1996
1996
1
$begingroup$
Okay, it seems to be equivalent to the formulation: “The copowering is a monoidal functor with respect to the component-wise monoidal structure.”
$endgroup$
– FKranhold
1 hour ago
add a comment |
1
$begingroup$
Okay, it seems to be equivalent to the formulation: “The copowering is a monoidal functor with respect to the component-wise monoidal structure.”
$endgroup$
– FKranhold
1 hour ago
1
1
$begingroup$
Okay, it seems to be equivalent to the formulation: “The copowering is a monoidal functor with respect to the component-wise monoidal structure.”
$endgroup$
– FKranhold
1 hour ago
$begingroup$
Okay, it seems to be equivalent to the formulation: “The copowering is a monoidal functor with respect to the component-wise monoidal structure.”
$endgroup$
– FKranhold
1 hour ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No. Consider the case where $(M,otimes,1)$ is $(mathbf{Set},times,1)$, so the enrichment is vacuous, and $(C,oplus,0)$ is $(mathbf{Set},+,0)$, with copowering $odot$ given by $times$.
Then the morphism you ask for would give a map
$$(A times X) + (B times Y) longrightarrow (A times B) times (X + Y) $$
which doesn’t exist in general: consider $A = X = Y = 1$, $B = 0$.
However, there is a natural map in the other direction. There are natural maps $A to C(X,A odot X)$ and $B to C(Y,B odot Y)$, the structure maps of the copowering. Also, the definition of enriched monoidal category includes the condition that $oplus$ is an enriched bifunctor, so there’s a general map $C(X,X') otimes C(Y,Y') to C(X oplus Y, X' oplus Y')$. Putting these together, we get a map
$$ A otimes B longrightarrow C(X, A odot X) otimes C(Y, B odot Y) longrightarrow C(X oplus Y, (A odot X) oplus (B odot Y)) $$
which corresponds under copowering to a map $(A otimes B) odot (X oplus Y) to (A odot X) oplus (B odot Y)$.
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
$endgroup$
$begingroup$
Good counterexample! Then maybe there is a canonical morphism in the other direction? Otherwise, I have the above problem with the associativity axiom for algebras over operads, as long as we do not assume that the copowering is a monoidal functor …
$endgroup$
– FKranhold
1 hour ago
$begingroup$
@FKranhold: Yes, there is a natural map in the converse direction — I’ll add the description of that in my answer.
$endgroup$
– Peter LeFanu Lumsdaine
16 mins ago
add a comment |
Your Answer
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1 Answer
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$begingroup$
No. Consider the case where $(M,otimes,1)$ is $(mathbf{Set},times,1)$, so the enrichment is vacuous, and $(C,oplus,0)$ is $(mathbf{Set},+,0)$, with copowering $odot$ given by $times$.
Then the morphism you ask for would give a map
$$(A times X) + (B times Y) longrightarrow (A times B) times (X + Y) $$
which doesn’t exist in general: consider $A = X = Y = 1$, $B = 0$.
However, there is a natural map in the other direction. There are natural maps $A to C(X,A odot X)$ and $B to C(Y,B odot Y)$, the structure maps of the copowering. Also, the definition of enriched monoidal category includes the condition that $oplus$ is an enriched bifunctor, so there’s a general map $C(X,X') otimes C(Y,Y') to C(X oplus Y, X' oplus Y')$. Putting these together, we get a map
$$ A otimes B longrightarrow C(X, A odot X) otimes C(Y, B odot Y) longrightarrow C(X oplus Y, (A odot X) oplus (B odot Y)) $$
which corresponds under copowering to a map $(A otimes B) odot (X oplus Y) to (A odot X) oplus (B odot Y)$.
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
$endgroup$
$begingroup$
Good counterexample! Then maybe there is a canonical morphism in the other direction? Otherwise, I have the above problem with the associativity axiom for algebras over operads, as long as we do not assume that the copowering is a monoidal functor …
$endgroup$
– FKranhold
1 hour ago
$begingroup$
@FKranhold: Yes, there is a natural map in the converse direction — I’ll add the description of that in my answer.
$endgroup$
– Peter LeFanu Lumsdaine
16 mins ago
add a comment |
$begingroup$
No. Consider the case where $(M,otimes,1)$ is $(mathbf{Set},times,1)$, so the enrichment is vacuous, and $(C,oplus,0)$ is $(mathbf{Set},+,0)$, with copowering $odot$ given by $times$.
Then the morphism you ask for would give a map
$$(A times X) + (B times Y) longrightarrow (A times B) times (X + Y) $$
which doesn’t exist in general: consider $A = X = Y = 1$, $B = 0$.
However, there is a natural map in the other direction. There are natural maps $A to C(X,A odot X)$ and $B to C(Y,B odot Y)$, the structure maps of the copowering. Also, the definition of enriched monoidal category includes the condition that $oplus$ is an enriched bifunctor, so there’s a general map $C(X,X') otimes C(Y,Y') to C(X oplus Y, X' oplus Y')$. Putting these together, we get a map
$$ A otimes B longrightarrow C(X, A odot X) otimes C(Y, B odot Y) longrightarrow C(X oplus Y, (A odot X) oplus (B odot Y)) $$
which corresponds under copowering to a map $(A otimes B) odot (X oplus Y) to (A odot X) oplus (B odot Y)$.
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
$endgroup$
$begingroup$
Good counterexample! Then maybe there is a canonical morphism in the other direction? Otherwise, I have the above problem with the associativity axiom for algebras over operads, as long as we do not assume that the copowering is a monoidal functor …
$endgroup$
– FKranhold
1 hour ago
$begingroup$
@FKranhold: Yes, there is a natural map in the converse direction — I’ll add the description of that in my answer.
$endgroup$
– Peter LeFanu Lumsdaine
16 mins ago
add a comment |
$begingroup$
No. Consider the case where $(M,otimes,1)$ is $(mathbf{Set},times,1)$, so the enrichment is vacuous, and $(C,oplus,0)$ is $(mathbf{Set},+,0)$, with copowering $odot$ given by $times$.
Then the morphism you ask for would give a map
$$(A times X) + (B times Y) longrightarrow (A times B) times (X + Y) $$
which doesn’t exist in general: consider $A = X = Y = 1$, $B = 0$.
However, there is a natural map in the other direction. There are natural maps $A to C(X,A odot X)$ and $B to C(Y,B odot Y)$, the structure maps of the copowering. Also, the definition of enriched monoidal category includes the condition that $oplus$ is an enriched bifunctor, so there’s a general map $C(X,X') otimes C(Y,Y') to C(X oplus Y, X' oplus Y')$. Putting these together, we get a map
$$ A otimes B longrightarrow C(X, A odot X) otimes C(Y, B odot Y) longrightarrow C(X oplus Y, (A odot X) oplus (B odot Y)) $$
which corresponds under copowering to a map $(A otimes B) odot (X oplus Y) to (A odot X) oplus (B odot Y)$.
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
$endgroup$
No. Consider the case where $(M,otimes,1)$ is $(mathbf{Set},times,1)$, so the enrichment is vacuous, and $(C,oplus,0)$ is $(mathbf{Set},+,0)$, with copowering $odot$ given by $times$.
Then the morphism you ask for would give a map
$$(A times X) + (B times Y) longrightarrow (A times B) times (X + Y) $$
which doesn’t exist in general: consider $A = X = Y = 1$, $B = 0$.
However, there is a natural map in the other direction. There are natural maps $A to C(X,A odot X)$ and $B to C(Y,B odot Y)$, the structure maps of the copowering. Also, the definition of enriched monoidal category includes the condition that $oplus$ is an enriched bifunctor, so there’s a general map $C(X,X') otimes C(Y,Y') to C(X oplus Y, X' oplus Y')$. Putting these together, we get a map
$$ A otimes B longrightarrow C(X, A odot X) otimes C(Y, B odot Y) longrightarrow C(X oplus Y, (A odot X) oplus (B odot Y)) $$
which corresponds under copowering to a map $(A otimes B) odot (X oplus Y) to (A odot X) oplus (B odot Y)$.
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
edited 11 mins ago
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
answered 1 hour ago
Peter LeFanu LumsdainePeter LeFanu Lumsdaine
8,80613871
8,80613871
$begingroup$
Good counterexample! Then maybe there is a canonical morphism in the other direction? Otherwise, I have the above problem with the associativity axiom for algebras over operads, as long as we do not assume that the copowering is a monoidal functor …
$endgroup$
– FKranhold
1 hour ago
$begingroup$
@FKranhold: Yes, there is a natural map in the converse direction — I’ll add the description of that in my answer.
$endgroup$
– Peter LeFanu Lumsdaine
16 mins ago
add a comment |
$begingroup$
Good counterexample! Then maybe there is a canonical morphism in the other direction? Otherwise, I have the above problem with the associativity axiom for algebras over operads, as long as we do not assume that the copowering is a monoidal functor …
$endgroup$
– FKranhold
1 hour ago
$begingroup$
@FKranhold: Yes, there is a natural map in the converse direction — I’ll add the description of that in my answer.
$endgroup$
– Peter LeFanu Lumsdaine
16 mins ago
$begingroup$
Good counterexample! Then maybe there is a canonical morphism in the other direction? Otherwise, I have the above problem with the associativity axiom for algebras over operads, as long as we do not assume that the copowering is a monoidal functor …
$endgroup$
– FKranhold
1 hour ago
$begingroup$
Good counterexample! Then maybe there is a canonical morphism in the other direction? Otherwise, I have the above problem with the associativity axiom for algebras over operads, as long as we do not assume that the copowering is a monoidal functor …
$endgroup$
– FKranhold
1 hour ago
$begingroup$
@FKranhold: Yes, there is a natural map in the converse direction — I’ll add the description of that in my answer.
$endgroup$
– Peter LeFanu Lumsdaine
16 mins ago
$begingroup$
@FKranhold: Yes, there is a natural map in the converse direction — I’ll add the description of that in my answer.
$endgroup$
– Peter LeFanu Lumsdaine
16 mins ago
add a comment |
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$begingroup$
Okay, it seems to be equivalent to the formulation: “The copowering is a monoidal functor with respect to the component-wise monoidal structure.”
$endgroup$
– FKranhold
1 hour ago