Category of Multisets and Spans
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I have been thinking about multisets for a while. These are sets where elements can repeat, so $S ={ a,a,b,c,b}$ is a multiset on the set $A = {a,b,c}$.
I have also been looking into morphisms between multisets. Take two multisets $S_A, S_B$ with underlying sets $A, B$. I would like to define a morphism between multisets $S_A, S_B$ as a span on the underlying sets, so $f = A leftarrow C rightarrow B$, and $f: S_A rightarrow S_B$.
Here is how I am defining the morphisms. Take a multiset $S_A$ and let $a_i$ be one of the terms, likewise for $S_B$ and $b_j$. We have an indexing set $C$ and let the $f, g$ be arms of a span so $f: C rightarrow A$ and $g: C rightarrow B$. Let $c_i in C$ and let $f(c_i)$ be the set element of term $a_i$, likewise for $g(c_j)$. I am getting the feeling this is impossible, but it completely makes sense in terms of three columns in a database, $A, B, C$. I guess it doesn't make sense to talk about a "term" of a multiset and its corresponding "set element". The category of algebras for finite multisets is $mathbb{N}$-modules, so they are like vectors spaces where we have terms. You can define a a vector via a span between the set of basis elements and the field, likewise for a module with a semiring.
I am not sure how to define span composition. I want a category of multisets with morphisms as defined. Does there exist a composition that gives such a category? I realize there are many options for defining the composition, and I really don't know which one I want. Is there a body of work that focuses on what all the different choices of composition mean for this particular problem?
Does my definition of the objects and morphisms define a category?
category-theory multisets
asked Nov 14 at 22:01
Ben Sprott
427312
add a comment |
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1
down vote
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I have been thinking about multisets for a while. These are sets where elements can repeat, so $S ={ a,a,b,c,b}$ is a multiset on the set $A = {a,b,c}$.
I have also been looking into morphisms between multisets. Take two multisets $S_A, S_B$ with underlying sets $A, B$. I would like to define a morphism between multisets $S_A, S_B$ as a span on the underlying sets, so $f = A leftarrow C rightarrow B$, and $f: S_A rightarrow S_B$.
Here is how I am defining the morphisms. Take a multiset $S_A$ and let $a_i$ be one of the terms, likewise for $S_B$ and $b_j$. We have an indexing set $C$ and let the $f, g$ be arms of a span so $f: C rightarrow A$ and $g: C rightarrow B$. Let $c_i in C$ and let $f(c_i)$ be the set element of term $a_i$, likewise for $g(c_j)$. I am getting the feeling this is impossible, but it completely makes sense in terms of three columns in a database, $A, B, C$. I guess it doesn't make sense to talk about a "term" of a multiset and its corresponding "set element". The category of algebras for finite multisets is $mathbb{N}$-modules, so they are like vectors spaces where we have terms. You can define a a vector via a span between the set of basis elements and the field, likewise for a module with a semiring.
I am not sure how to define span composition. I want a category of multisets with morphisms as defined. Does there exist a composition that gives such a category? I realize there are many options for defining the composition, and I really don't know which one I want. Is there a body of work that focuses on what all the different choices of composition mean for this particular problem?
Does my definition of the objects and morphisms define a category?
category-theory multisets
asked Nov 14 at 22:01
Ben Sprott
427312
4
Your definition of the morphisms is bizarre: you're saying that $S_A$ and $S_B$ don't matter at all for defining a morphism $S_Ato S_B$, only the sets $A$ and $B$ matter??
– Eric Wofsey
Nov 14 at 22:59
@EricWofsey Hi, I tried to give my intuition on how the morphisms work. It is based on the idea that you can describe a vector via a span between basis elements and the field, or semiring multiplier.
– Ben Sprott
Nov 15 at 15:31
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have been thinking about multisets for a while. These are sets where elements can repeat, so $S ={ a,a,b,c,b}$ is a multiset on the set $A = {a,b,c}$.
I have also been looking into morphisms between multisets. Take two multisets $S_A, S_B$ with underlying sets $A, B$. I would like to define a morphism between multisets $S_A, S_B$ as a span on the underlying sets, so $f = A leftarrow C rightarrow B$, and $f: S_A rightarrow S_B$.
Here is how I am defining the morphisms. Take a multiset $S_A$ and let $a_i$ be one of the terms, likewise for $S_B$ and $b_j$. We have an indexing set $C$ and let the $f, g$ be arms of a span so $f: C rightarrow A$ and $g: C rightarrow B$. Let $c_i in C$ and let $f(c_i)$ be the set element of term $a_i$, likewise for $g(c_j)$. I am getting the feeling this is impossible, but it completely makes sense in terms of three columns in a database, $A, B, C$. I guess it doesn't make sense to talk about a "term" of a multiset and its corresponding "set element". The category of algebras for finite multisets is $mathbb{N}$-modules, so they are like vectors spaces where we have terms. You can define a a vector via a span between the set of basis elements and the field, likewise for a module with a semiring.
I am not sure how to define span composition. I want a category of multisets with morphisms as defined. Does there exist a composition that gives such a category? I realize there are many options for defining the composition, and I really don't know which one I want. Is there a body of work that focuses on what all the different choices of composition mean for this particular problem?
Does my definition of the objects and morphisms define a category?
category-theory multisets
asked Nov 14 at 22:01
Ben Sprott
427312
I have been thinking about multisets for a while. These are sets where elements can repeat, so $S ={ a,a,b,c,b}$ is a multiset on the set $A = {a,b,c}$.
I have also been looking into morphisms between multisets. Take two multisets $S_A, S_B$ with underlying sets $A, B$. I would like to define a morphism between multisets $S_A, S_B$ as a span on the underlying sets, so $f = A leftarrow C rightarrow B$, and $f: S_A rightarrow S_B$.
Here is how I am defining the morphisms. Take a multiset $S_A$ and let $a_i$ be one of the terms, likewise for $S_B$ and $b_j$. We have an indexing set $C$ and let the $f, g$ be arms of a span so $f: C rightarrow A$ and $g: C rightarrow B$. Let $c_i in C$ and let $f(c_i)$ be the set element of term $a_i$, likewise for $g(c_j)$. I am getting the feeling this is impossible, but it completely makes sense in terms of three columns in a database, $A, B, C$. I guess it doesn't make sense to talk about a "term" of a multiset and its corresponding "set element". The category of algebras for finite multisets is $mathbb{N}$-modules, so they are like vectors spaces where we have terms. You can define a a vector via a span between the set of basis elements and the field, likewise for a module with a semiring.
I am not sure how to define span composition. I want a category of multisets with morphisms as defined. Does there exist a composition that gives such a category? I realize there are many options for defining the composition, and I really don't know which one I want. Is there a body of work that focuses on what all the different choices of composition mean for this particular problem?
Does my definition of the objects and morphisms define a category?
category-theory multisets
category-theory multisets
asked Nov 14 at 22:01
Ben Sprott
427312
asked Nov 14 at 22:01
Ben Sprott
427312
edited 8 hours ago
asked Nov 14 at 22:01
Ben Sprott
427312
asked Nov 14 at 22:01
Ben Sprott
427312
asked Nov 14 at 22:01
Ben Sprott
427312
427312
4
Your definition of the morphisms is bizarre: you're saying that $S_A$ and $S_B$ don't matter at all for defining a morphism $S_Ato S_B$, only the sets $A$ and $B$ matter??
– Eric Wofsey
Nov 14 at 22:59
@EricWofsey Hi, I tried to give my intuition on how the morphisms work. It is based on the idea that you can describe a vector via a span between basis elements and the field, or semiring multiplier.
– Ben Sprott
Nov 15 at 15:31
add a comment |
4
Your definition of the morphisms is bizarre: you're saying that $S_A$ and $S_B$ don't matter at all for defining a morphism $S_Ato S_B$, only the sets $A$ and $B$ matter??
– Eric Wofsey
Nov 14 at 22:59
@EricWofsey Hi, I tried to give my intuition on how the morphisms work. It is based on the idea that you can describe a vector via a span between basis elements and the field, or semiring multiplier.
– Ben Sprott
Nov 15 at 15:31
4
4
Your definition of the morphisms is bizarre: you're saying that $S_A$ and $S_B$ don't matter at all for defining a morphism $S_Ato S_B$, only the sets $A$ and $B$ matter??
– Eric Wofsey
Nov 14 at 22:59
Your definition of the morphisms is bizarre: you're saying that $S_A$ and $S_B$ don't matter at all for defining a morphism $S_Ato S_B$, only the sets $A$ and $B$ matter??
– Eric Wofsey
Nov 14 at 22:59
@EricWofsey Hi, I tried to give my intuition on how the morphisms work. It is based on the idea that you can describe a vector via a span between basis elements and the field, or semiring multiplier.
– Ben Sprott
Nov 15 at 15:31
@EricWofsey Hi, I tried to give my intuition on how the morphisms work. It is based on the idea that you can describe a vector via a span between basis elements and the field, or semiring multiplier.
– Ben Sprott
Nov 15 at 15:31
add a comment |
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4
Your definition of the morphisms is bizarre: you're saying that $S_A$ and $S_B$ don't matter at all for defining a morphism $S_Ato S_B$, only the sets $A$ and $B$ matter??
– Eric Wofsey
Nov 14 at 22:59
@EricWofsey Hi, I tried to give my intuition on how the morphisms work. It is based on the idea that you can describe a vector via a span between basis elements and the field, or semiring multiplier.
– Ben Sprott
Nov 15 at 15:31