Direct Sum of Additive Categories
$begingroup$
I am following Etingof et al's book on tensor categories. They define the direct sum of additive categories as follows (I am rephrasing):
Let ${ mathcal C_alpha, alpha in I }$ be a family of additive categories.
The direct sum $mathcal C=underset{alphain I}{bigoplus}{mathcal C}_alpha$ is the category whose objects are
$$X = underset{alphain I}{bigoplus}X_alpha ,quad X_alpha in mathcal C_alpha$$
such that only a finite number of $X_alpha$ are $neq 0_alpha$. The arrows are
$$
text{Hom}_mathcal C(underset{alphain I}{bigoplus}X_alpha,underset{alphain I}{bigoplus}Y_alpha) = underset{alphain I}{bigoplus} text{Hom}_{mathcal C_alpha}(X_alpha,Y_alpha)
$$
My problem with this: how does it make sense to do the direct sum of objects of different additive categories? This definition hinges on this possibility, but we only know that we have the direct sum of any two objects in the same additive category, not different categories...
category-theory additive-categories
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
$endgroup$
add a comment |
$begingroup$
I am following Etingof et al's book on tensor categories. They define the direct sum of additive categories as follows (I am rephrasing):
Let ${ mathcal C_alpha, alpha in I }$ be a family of additive categories.
The direct sum $mathcal C=underset{alphain I}{bigoplus}{mathcal C}_alpha$ is the category whose objects are
$$X = underset{alphain I}{bigoplus}X_alpha ,quad X_alpha in mathcal C_alpha$$
such that only a finite number of $X_alpha$ are $neq 0_alpha$. The arrows are
$$
text{Hom}_mathcal C(underset{alphain I}{bigoplus}X_alpha,underset{alphain I}{bigoplus}Y_alpha) = underset{alphain I}{bigoplus} text{Hom}_{mathcal C_alpha}(X_alpha,Y_alpha)
$$
My problem with this: how does it make sense to do the direct sum of objects of different additive categories? This definition hinges on this possibility, but we only know that we have the direct sum of any two objects in the same additive category, not different categories...
category-theory additive-categories
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
$endgroup$
3
$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in: An object in the direct sum category is an $I$-indexed list of objects, one from each category, all but finitely many nonzero, and it’s denoted by this suggestive direct sum notation. As for the Hom sets, these are actually abelian groups so you can sum them.
$endgroup$
– Ben
Dec 9 '18 at 15:09
add a comment |
$begingroup$
I am following Etingof et al's book on tensor categories. They define the direct sum of additive categories as follows (I am rephrasing):
Let ${ mathcal C_alpha, alpha in I }$ be a family of additive categories.
The direct sum $mathcal C=underset{alphain I}{bigoplus}{mathcal C}_alpha$ is the category whose objects are
$$X = underset{alphain I}{bigoplus}X_alpha ,quad X_alpha in mathcal C_alpha$$
such that only a finite number of $X_alpha$ are $neq 0_alpha$. The arrows are
$$
text{Hom}_mathcal C(underset{alphain I}{bigoplus}X_alpha,underset{alphain I}{bigoplus}Y_alpha) = underset{alphain I}{bigoplus} text{Hom}_{mathcal C_alpha}(X_alpha,Y_alpha)
$$
My problem with this: how does it make sense to do the direct sum of objects of different additive categories? This definition hinges on this possibility, but we only know that we have the direct sum of any two objects in the same additive category, not different categories...
category-theory additive-categories
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
$endgroup$
I am following Etingof et al's book on tensor categories. They define the direct sum of additive categories as follows (I am rephrasing):
Let ${ mathcal C_alpha, alpha in I }$ be a family of additive categories.
The direct sum $mathcal C=underset{alphain I}{bigoplus}{mathcal C}_alpha$ is the category whose objects are
$$X = underset{alphain I}{bigoplus}X_alpha ,quad X_alpha in mathcal C_alpha$$
such that only a finite number of $X_alpha$ are $neq 0_alpha$. The arrows are
$$
text{Hom}_mathcal C(underset{alphain I}{bigoplus}X_alpha,underset{alphain I}{bigoplus}Y_alpha) = underset{alphain I}{bigoplus} text{Hom}_{mathcal C_alpha}(X_alpha,Y_alpha)
$$
My problem with this: how does it make sense to do the direct sum of objects of different additive categories? This definition hinges on this possibility, but we only know that we have the direct sum of any two objects in the same additive category, not different categories...
category-theory additive-categories
category-theory additive-categories
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
asked Dec 9 '18 at 14:58
SoapSoap
1,022615
1,022615
3
$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in: An object in the direct sum category is an $I$-indexed list of objects, one from each category, all but finitely many nonzero, and it’s denoted by this suggestive direct sum notation. As for the Hom sets, these are actually abelian groups so you can sum them.
$endgroup$
– Ben
Dec 9 '18 at 15:09
add a comment |
3
$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in: An object in the direct sum category is an $I$-indexed list of objects, one from each category, all but finitely many nonzero, and it’s denoted by this suggestive direct sum notation. As for the Hom sets, these are actually abelian groups so you can sum them.
$endgroup$
– Ben
Dec 9 '18 at 15:09
3
3
$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in: An object in the direct sum category is an $I$-indexed list of objects, one from each category, all but finitely many nonzero, and it’s denoted by this suggestive direct sum notation. As for the Hom sets, these are actually abelian groups so you can sum them.
$endgroup$
– Ben
Dec 9 '18 at 15:09
$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in: An object in the direct sum category is an $I$-indexed list of objects, one from each category, all but finitely many nonzero, and it’s denoted by this suggestive direct sum notation. As for the Hom sets, these are actually abelian groups so you can sum them.
$endgroup$
– Ben
Dec 9 '18 at 15:09
add a comment |
1 Answer
1
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$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in:
An object $X$ in the direct sum category is an $I$-indexed list of objects $X_i$, one from each category, finitely many nonzero, and will be denoted by the suggestive notation $X = bigoplus_{i in I} X_i$. As for the $Hom$ sets, these are actually abelian groups, so you can sum them.
If this notation needed any more justification, note that once you've built this category $mathcal C = bigoplus mathcal C_i$, there are then embeddings $F_i: mathcal C_i to mathcal C$ of additive categories for which one actually has a natural identification $$bigoplus F_i(X_i) = bigoplus X_i$$ where on the left the $bigoplus$ is actually a direct sum in an additive category and on the right the $bigoplus$ is the "suggestive notation".
Continuing this discussion for what is now much longer than necessary, you could compare this situation to taking the external direct sum of vector spaces $V_i$. Given $v_iin V_i$ of course it is illegal to form $sum v_i$. But once you define $bigoplus V_i$ and identify the $V_i$ as subspaces of it, it becomes "actually" a sum in some vector space.
answered Dec 10 '18 at 11:16
BenBen
3,851616
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add a comment |
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$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in:
An object $X$ in the direct sum category is an $I$-indexed list of objects $X_i$, one from each category, finitely many nonzero, and will be denoted by the suggestive notation $X = bigoplus_{i in I} X_i$. As for the $Hom$ sets, these are actually abelian groups, so you can sum them.
If this notation needed any more justification, note that once you've built this category $mathcal C = bigoplus mathcal C_i$, there are then embeddings $F_i: mathcal C_i to mathcal C$ of additive categories for which one actually has a natural identification $$bigoplus F_i(X_i) = bigoplus X_i$$ where on the left the $bigoplus$ is actually a direct sum in an additive category and on the right the $bigoplus$ is the "suggestive notation".
Continuing this discussion for what is now much longer than necessary, you could compare this situation to taking the external direct sum of vector spaces $V_i$. Given $v_iin V_i$ of course it is illegal to form $sum v_i$. But once you define $bigoplus V_i$ and identify the $V_i$ as subspaces of it, it becomes "actually" a sum in some vector space.
answered Dec 10 '18 at 11:16
BenBen
3,851616
$endgroup$
add a comment |
$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in:
An object $X$ in the direct sum category is an $I$-indexed list of objects $X_i$, one from each category, finitely many nonzero, and will be denoted by the suggestive notation $X = bigoplus_{i in I} X_i$. As for the $Hom$ sets, these are actually abelian groups, so you can sum them.
If this notation needed any more justification, note that once you've built this category $mathcal C = bigoplus mathcal C_i$, there are then embeddings $F_i: mathcal C_i to mathcal C$ of additive categories for which one actually has a natural identification $$bigoplus F_i(X_i) = bigoplus X_i$$ where on the left the $bigoplus$ is actually a direct sum in an additive category and on the right the $bigoplus$ is the "suggestive notation".
Continuing this discussion for what is now much longer than necessary, you could compare this situation to taking the external direct sum of vector spaces $V_i$. Given $v_iin V_i$ of course it is illegal to form $sum v_i$. But once you define $bigoplus V_i$ and identify the $V_i$ as subspaces of it, it becomes "actually" a sum in some vector space.
answered Dec 10 '18 at 11:16
BenBen
3,851616
$endgroup$
add a comment |
$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in:
An object $X$ in the direct sum category is an $I$-indexed list of objects $X_i$, one from each category, finitely many nonzero, and will be denoted by the suggestive notation $X = bigoplus_{i in I} X_i$. As for the $Hom$ sets, these are actually abelian groups, so you can sum them.
If this notation needed any more justification, note that once you've built this category $mathcal C = bigoplus mathcal C_i$, there are then embeddings $F_i: mathcal C_i to mathcal C$ of additive categories for which one actually has a natural identification $$bigoplus F_i(X_i) = bigoplus X_i$$ where on the left the $bigoplus$ is actually a direct sum in an additive category and on the right the $bigoplus$ is the "suggestive notation".
Continuing this discussion for what is now much longer than necessary, you could compare this situation to taking the external direct sum of vector spaces $V_i$. Given $v_iin V_i$ of course it is illegal to form $sum v_i$. But once you define $bigoplus V_i$ and identify the $V_i$ as subspaces of it, it becomes "actually" a sum in some vector space.
answered Dec 10 '18 at 11:16
BenBen
3,851616
$endgroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in:
An object $X$ in the direct sum category is an $I$-indexed list of objects $X_i$, one from each category, finitely many nonzero, and will be denoted by the suggestive notation $X = bigoplus_{i in I} X_i$. As for the $Hom$ sets, these are actually abelian groups, so you can sum them.
If this notation needed any more justification, note that once you've built this category $mathcal C = bigoplus mathcal C_i$, there are then embeddings $F_i: mathcal C_i to mathcal C$ of additive categories for which one actually has a natural identification $$bigoplus F_i(X_i) = bigoplus X_i$$ where on the left the $bigoplus$ is actually a direct sum in an additive category and on the right the $bigoplus$ is the "suggestive notation".
Continuing this discussion for what is now much longer than necessary, you could compare this situation to taking the external direct sum of vector spaces $V_i$. Given $v_iin V_i$ of course it is illegal to form $sum v_i$. But once you define $bigoplus V_i$ and identify the $V_i$ as subspaces of it, it becomes "actually" a sum in some vector space.
answered Dec 10 '18 at 11:16
BenBen
3,851616
answered Dec 10 '18 at 11:16
BenBen
3,851616
answered Dec 10 '18 at 11:16
BenBen
3,851616
answered Dec 10 '18 at 11:16
BenBen
3,851616
3,851616
add a comment |
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$begingroup$
I didn’t read their book but presumably this is just a notation/definition combo, as in: An object in the direct sum category is an $I$-indexed list of objects, one from each category, all but finitely many nonzero, and it’s denoted by this suggestive direct sum notation. As for the Hom sets, these are actually abelian groups so you can sum them.
$endgroup$
– Ben
Dec 9 '18 at 15:09