Formalizing the idea of this “equivalence of data” in the category of vector spaces, and how does this generalize to other categories?Various definitions of group actionCategory Theory: homset preserves limitsInternalising the functor action on morphisms (e.g. to exponential objects)categorical generalizations of familiar objectsCategory equivalence of sets and vector spacesContinuations vs. YonedaWhat is the map $mathrmNat(F_1,F_2)timesmathrmNat(G_1,G_2)tomathrmNat(F_1circ G_1,F_2circ G_2)$?Ordinals in category theory?Why do exponential objects (in category theory) require currying?How to prove that $V otimes W cong V^ast ast otimes W$?
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Formalizing the idea of this “equivalence of data” in the category of vector spaces, and how does this generalize to other categories?
Various definitions of group actionCategory Theory: homset preserves limitsInternalising the functor action on morphisms (e.g. to exponential objects)categorical generalizations of familiar objectsCategory equivalence of sets and vector spacesContinuations vs. YonedaWhat is the map $mathrmNat(F_1,F_2)timesmathrmNat(G_1,G_2)tomathrmNat(F_1circ G_1,F_2circ G_2)$?Ordinals in category theory?Why do exponential objects (in category theory) require currying?How to prove that $V otimes W cong V^ast ast otimes W$?
$begingroup$
My category theory is almost nonexistent, but this seems like a "categorical idea". So I'm looking to formalize this idea: Given the data of $U,V,W$ vector spaces the following are "equivalent"
$$B:Uto Hom(V,W)$$
where $B$ is a linear map and
$$B:Utimes V to W$$
where $B$ is a bilinear map. Assuming the top, we can define our bottom map as
$$(u,v)mapsto B(u)(v)$$
and this is linear in $U$ since $B$ up top is a linear map, and linear in $V$ since $B(u)(-)$ is a linear map. Now assuming the bottom, we can just "freeze" the V component. So fix arbitrary $vin V$, then
$$B:Uto Hom(V,W)$$
$$umapsto B((u,v))$$
So yeah, how do I formalize this notion? And how far does this idea generalize?
linear-algebra category-theory
asked Mar 14 at 17:42
Paul TPaul T
636
$endgroup$
|
show 1 more comment
$begingroup$
My category theory is almost nonexistent, but this seems like a "categorical idea". So I'm looking to formalize this idea: Given the data of $U,V,W$ vector spaces the following are "equivalent"
$$B:Uto Hom(V,W)$$
where $B$ is a linear map and
$$B:Utimes V to W$$
where $B$ is a bilinear map. Assuming the top, we can define our bottom map as
$$(u,v)mapsto B(u)(v)$$
and this is linear in $U$ since $B$ up top is a linear map, and linear in $V$ since $B(u)(-)$ is a linear map. Now assuming the bottom, we can just "freeze" the V component. So fix arbitrary $vin V$, then
$$B:Uto Hom(V,W)$$
$$umapsto B((u,v))$$
So yeah, how do I formalize this notion? And how far does this idea generalize?
linear-algebra category-theory
asked Mar 14 at 17:42
Paul TPaul T
636
$endgroup$
1
$begingroup$
You need a cartesian closed category: a category that has products and exponentials. The adjointness between products and exponentials (also called currying), I think, is what you're looking for.
$endgroup$
– frabala
Mar 14 at 17:52
$begingroup$
thanks! this is what i wanted
$endgroup$
– Paul T
Mar 14 at 18:10
4
$begingroup$
frabala is slightly incorrect. See ncatlab.org/nlab/show/closed+monoidal+category
$endgroup$
– Qiaochu Yuan
Mar 14 at 18:55
$begingroup$
Have you read about tensor products, and the tensor-hom adjunction?
$endgroup$
– Joppy
Mar 15 at 13:29
$begingroup$
About the title: "data" like what you suggest is often encoded as a functor on a category, so more generally, equivalence of data can be seen as a natural isomorphism of functors. Here your data is on the one hand $hom(U,hom(V,W))$ and on the other $Bil(U,V;W)$ : that they are equivalent "data" can be seen by the fact that they are isomorphic functors. In this special case, as $Bil(U,V;W) simeq hom(Uotimes V, W)$, this natural isomorphism of functors is actually a special case of an adjunction.
$endgroup$
– Max
Mar 15 at 18:15
|
show 1 more comment
$begingroup$
My category theory is almost nonexistent, but this seems like a "categorical idea". So I'm looking to formalize this idea: Given the data of $U,V,W$ vector spaces the following are "equivalent"
$$B:Uto Hom(V,W)$$
where $B$ is a linear map and
$$B:Utimes V to W$$
where $B$ is a bilinear map. Assuming the top, we can define our bottom map as
$$(u,v)mapsto B(u)(v)$$
and this is linear in $U$ since $B$ up top is a linear map, and linear in $V$ since $B(u)(-)$ is a linear map. Now assuming the bottom, we can just "freeze" the V component. So fix arbitrary $vin V$, then
$$B:Uto Hom(V,W)$$
$$umapsto B((u,v))$$
So yeah, how do I formalize this notion? And how far does this idea generalize?
linear-algebra category-theory
asked Mar 14 at 17:42
Paul TPaul T
636
$endgroup$
My category theory is almost nonexistent, but this seems like a "categorical idea". So I'm looking to formalize this idea: Given the data of $U,V,W$ vector spaces the following are "equivalent"
$$B:Uto Hom(V,W)$$
where $B$ is a linear map and
$$B:Utimes V to W$$
where $B$ is a bilinear map. Assuming the top, we can define our bottom map as
$$(u,v)mapsto B(u)(v)$$
and this is linear in $U$ since $B$ up top is a linear map, and linear in $V$ since $B(u)(-)$ is a linear map. Now assuming the bottom, we can just "freeze" the V component. So fix arbitrary $vin V$, then
$$B:Uto Hom(V,W)$$
$$umapsto B((u,v))$$
So yeah, how do I formalize this notion? And how far does this idea generalize?
linear-algebra category-theory
linear-algebra category-theory
asked Mar 14 at 17:42
Paul TPaul T
636
asked Mar 14 at 17:42
Paul TPaul T
636
asked Mar 14 at 17:42
Paul TPaul T
636
asked Mar 14 at 17:42
Paul TPaul T
636
asked Mar 14 at 17:42
Paul TPaul T
636
636
1
$begingroup$
You need a cartesian closed category: a category that has products and exponentials. The adjointness between products and exponentials (also called currying), I think, is what you're looking for.
$endgroup$
– frabala
Mar 14 at 17:52
$begingroup$
thanks! this is what i wanted
$endgroup$
– Paul T
Mar 14 at 18:10
4
$begingroup$
frabala is slightly incorrect. See ncatlab.org/nlab/show/closed+monoidal+category
$endgroup$
– Qiaochu Yuan
Mar 14 at 18:55
$begingroup$
Have you read about tensor products, and the tensor-hom adjunction?
$endgroup$
– Joppy
Mar 15 at 13:29
$begingroup$
About the title: "data" like what you suggest is often encoded as a functor on a category, so more generally, equivalence of data can be seen as a natural isomorphism of functors. Here your data is on the one hand $hom(U,hom(V,W))$ and on the other $Bil(U,V;W)$ : that they are equivalent "data" can be seen by the fact that they are isomorphic functors. In this special case, as $Bil(U,V;W) simeq hom(Uotimes V, W)$, this natural isomorphism of functors is actually a special case of an adjunction.
$endgroup$
– Max
Mar 15 at 18:15
|
show 1 more comment
1
$begingroup$
You need a cartesian closed category: a category that has products and exponentials. The adjointness between products and exponentials (also called currying), I think, is what you're looking for.
$endgroup$
– frabala
Mar 14 at 17:52
$begingroup$
thanks! this is what i wanted
$endgroup$
– Paul T
Mar 14 at 18:10
4
$begingroup$
frabala is slightly incorrect. See ncatlab.org/nlab/show/closed+monoidal+category
$endgroup$
– Qiaochu Yuan
Mar 14 at 18:55
$begingroup$
Have you read about tensor products, and the tensor-hom adjunction?
$endgroup$
– Joppy
Mar 15 at 13:29
$begingroup$
About the title: "data" like what you suggest is often encoded as a functor on a category, so more generally, equivalence of data can be seen as a natural isomorphism of functors. Here your data is on the one hand $hom(U,hom(V,W))$ and on the other $Bil(U,V;W)$ : that they are equivalent "data" can be seen by the fact that they are isomorphic functors. In this special case, as $Bil(U,V;W) simeq hom(Uotimes V, W)$, this natural isomorphism of functors is actually a special case of an adjunction.
$endgroup$
– Max
Mar 15 at 18:15
1
1
$begingroup$
You need a cartesian closed category: a category that has products and exponentials. The adjointness between products and exponentials (also called currying), I think, is what you're looking for.
$endgroup$
– frabala
Mar 14 at 17:52
$begingroup$
You need a cartesian closed category: a category that has products and exponentials. The adjointness between products and exponentials (also called currying), I think, is what you're looking for.
$endgroup$
– frabala
Mar 14 at 17:52
$begingroup$
thanks! this is what i wanted
$endgroup$
– Paul T
Mar 14 at 18:10
$begingroup$
thanks! this is what i wanted
$endgroup$
– Paul T
Mar 14 at 18:10
4
4
$begingroup$
frabala is slightly incorrect. See ncatlab.org/nlab/show/closed+monoidal+category
$endgroup$
– Qiaochu Yuan
Mar 14 at 18:55
$begingroup$
frabala is slightly incorrect. See ncatlab.org/nlab/show/closed+monoidal+category
$endgroup$
– Qiaochu Yuan
Mar 14 at 18:55
$begingroup$
Have you read about tensor products, and the tensor-hom adjunction?
$endgroup$
– Joppy
Mar 15 at 13:29
$begingroup$
Have you read about tensor products, and the tensor-hom adjunction?
$endgroup$
– Joppy
Mar 15 at 13:29
$begingroup$
About the title: "data" like what you suggest is often encoded as a functor on a category, so more generally, equivalence of data can be seen as a natural isomorphism of functors. Here your data is on the one hand $hom(U,hom(V,W))$ and on the other $Bil(U,V;W)$ : that they are equivalent "data" can be seen by the fact that they are isomorphic functors. In this special case, as $Bil(U,V;W) simeq hom(Uotimes V, W)$, this natural isomorphism of functors is actually a special case of an adjunction.
$endgroup$
– Max
Mar 15 at 18:15
$begingroup$
About the title: "data" like what you suggest is often encoded as a functor on a category, so more generally, equivalence of data can be seen as a natural isomorphism of functors. Here your data is on the one hand $hom(U,hom(V,W))$ and on the other $Bil(U,V;W)$ : that they are equivalent "data" can be seen by the fact that they are isomorphic functors. In this special case, as $Bil(U,V;W) simeq hom(Uotimes V, W)$, this natural isomorphism of functors is actually a special case of an adjunction.
$endgroup$
– Max
Mar 15 at 18:15
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
$newcommandVecoperatornameVecnewcommandSetoperatornameSetnewcommandHomoperatornameHom$In the case of vector spaces (or, more generally, modules over a ring $R$) we have an adjunction
$$Votimes_k-:Vec_krightleftarrowsVec_k:Hom_k(V,-)$$
Thus for every vector spaces $U,W$ we have an isomorphism
$$Hom_k(Votimes_kU,W)congHom_k(U,Hom_k(V,W))$$
which formalize the corrispondece you give in the OP.
A similar adjunction holds, for example in category of sets, with the adjunction
$$Vtimes-:SetrightleftarrowsSet:(-)^V$$
which gives for every set $U,W$ a bijection
$$Hom(Vtimes U,W)congHom(U,W^V)$$
which can be written as
$$W^Vtimes Ucong (W^V)^U$$
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
$endgroup$
add a comment |
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$begingroup$
$newcommandVecoperatornameVecnewcommandSetoperatornameSetnewcommandHomoperatornameHom$In the case of vector spaces (or, more generally, modules over a ring $R$) we have an adjunction
$$Votimes_k-:Vec_krightleftarrowsVec_k:Hom_k(V,-)$$
Thus for every vector spaces $U,W$ we have an isomorphism
$$Hom_k(Votimes_kU,W)congHom_k(U,Hom_k(V,W))$$
which formalize the corrispondece you give in the OP.
A similar adjunction holds, for example in category of sets, with the adjunction
$$Vtimes-:SetrightleftarrowsSet:(-)^V$$
which gives for every set $U,W$ a bijection
$$Hom(Vtimes U,W)congHom(U,W^V)$$
which can be written as
$$W^Vtimes Ucong (W^V)^U$$
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
$endgroup$
add a comment |
$begingroup$
$newcommandVecoperatornameVecnewcommandSetoperatornameSetnewcommandHomoperatornameHom$In the case of vector spaces (or, more generally, modules over a ring $R$) we have an adjunction
$$Votimes_k-:Vec_krightleftarrowsVec_k:Hom_k(V,-)$$
Thus for every vector spaces $U,W$ we have an isomorphism
$$Hom_k(Votimes_kU,W)congHom_k(U,Hom_k(V,W))$$
which formalize the corrispondece you give in the OP.
A similar adjunction holds, for example in category of sets, with the adjunction
$$Vtimes-:SetrightleftarrowsSet:(-)^V$$
which gives for every set $U,W$ a bijection
$$Hom(Vtimes U,W)congHom(U,W^V)$$
which can be written as
$$W^Vtimes Ucong (W^V)^U$$
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
$endgroup$
add a comment |
$begingroup$
$newcommandVecoperatornameVecnewcommandSetoperatornameSetnewcommandHomoperatornameHom$In the case of vector spaces (or, more generally, modules over a ring $R$) we have an adjunction
$$Votimes_k-:Vec_krightleftarrowsVec_k:Hom_k(V,-)$$
Thus for every vector spaces $U,W$ we have an isomorphism
$$Hom_k(Votimes_kU,W)congHom_k(U,Hom_k(V,W))$$
which formalize the corrispondece you give in the OP.
A similar adjunction holds, for example in category of sets, with the adjunction
$$Vtimes-:SetrightleftarrowsSet:(-)^V$$
which gives for every set $U,W$ a bijection
$$Hom(Vtimes U,W)congHom(U,W^V)$$
which can be written as
$$W^Vtimes Ucong (W^V)^U$$
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
$endgroup$
$newcommandVecoperatornameVecnewcommandSetoperatornameSetnewcommandHomoperatornameHom$In the case of vector spaces (or, more generally, modules over a ring $R$) we have an adjunction
$$Votimes_k-:Vec_krightleftarrowsVec_k:Hom_k(V,-)$$
Thus for every vector spaces $U,W$ we have an isomorphism
$$Hom_k(Votimes_kU,W)congHom_k(U,Hom_k(V,W))$$
which formalize the corrispondece you give in the OP.
A similar adjunction holds, for example in category of sets, with the adjunction
$$Vtimes-:SetrightleftarrowsSet:(-)^V$$
which gives for every set $U,W$ a bijection
$$Hom(Vtimes U,W)congHom(U,W^V)$$
which can be written as
$$W^Vtimes Ucong (W^V)^U$$
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
answered Mar 15 at 12:12
Fabio LucchiniFabio Lucchini
9,34111426
9,34111426
add a comment |
add a comment |
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$begingroup$
You need a cartesian closed category: a category that has products and exponentials. The adjointness between products and exponentials (also called currying), I think, is what you're looking for.
$endgroup$
– frabala
Mar 14 at 17:52
$begingroup$
thanks! this is what i wanted
$endgroup$
– Paul T
Mar 14 at 18:10
4
$begingroup$
frabala is slightly incorrect. See ncatlab.org/nlab/show/closed+monoidal+category
$endgroup$
– Qiaochu Yuan
Mar 14 at 18:55
$begingroup$
Have you read about tensor products, and the tensor-hom adjunction?
$endgroup$
– Joppy
Mar 15 at 13:29
$begingroup$
About the title: "data" like what you suggest is often encoded as a functor on a category, so more generally, equivalence of data can be seen as a natural isomorphism of functors. Here your data is on the one hand $hom(U,hom(V,W))$ and on the other $Bil(U,V;W)$ : that they are equivalent "data" can be seen by the fact that they are isomorphic functors. In this special case, as $Bil(U,V;W) simeq hom(Uotimes V, W)$, this natural isomorphism of functors is actually a special case of an adjunction.
$endgroup$
– Max
Mar 15 at 18:15