Skip to main content
MathOverflow

Return to Question

Notice removed Improve details by Ben Sprott
Bounty Ended with Neil Strickland's answer chosen by Ben Sprott
Notice added Improve details by Ben Sprott
Bounty Started worth 50 reputation by Ben Sprott
added 277 characters in body
Source Link
Ben Sprott
Ben Sprott
  • 1.3k
  • 14
  • 23

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 $f: S_A \rightarrow S_B$. I am not sure how to define span composition. I am thinking a lot about David Spivak's work these days, so I am going to suggest a definition from his text, see section 2.5.2.3, which I cannot reproduce here. He uses fiber products.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?

I have only an intuition that tells me that this definition of morphism is both a morphism of multisets in the category defined but is also a tensor product. We know that the Eilenberg-Moore category for the multiset monad is actually $\mathbb{N}$-modules. We should find a notion of tensor product there. I am guessing that we can define a monad on Set that maps a set to its set of multisets and spans as defined. There should be a similar category of modules, and thus a tensor product. I can only guess that what I am thinking means that the spans, as defined, map to both morphisms in the category of modules and the tensor product in the category of modules.

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 $f: S_A \rightarrow S_B$. I am not sure how to define span composition. I am thinking a lot about David Spivak's work these days, so I am going to suggest a definition from his text, see section 2.5.2.3, which I cannot reproduce here. He uses fiber products.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?

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 $f: S_A \rightarrow S_B$. I am not sure how to define span composition. I am thinking a lot about David Spivak's work these days, so I am going to suggest a definition from his text, see section 2.5.2.3, which I cannot reproduce here. He uses fiber products.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?

I have only an intuition that tells me that this definition of morphism is both a morphism of multisets in the category defined but is also a tensor product. We know that the Eilenberg-Moore category for the multiset monad is actually $\mathbb{N}$-modules. We should find a notion of tensor product there. I am guessing that we can define a monad on Set that maps a set to its set of multisets and spans as defined. There should be a similar category of modules, and thus a tensor product. I can only guess that what I am thinking means that the spans, as defined, map to both morphisms in the category of modules and the tensor product in the category of modules.

added 277 characters in body
Source Link
Ben Sprott
Ben Sprott
  • 1.3k
  • 14
  • 23

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 $f: S_A \rightarrow S_B$. I am not sure how to define span composition. I am thinking a lot about David Spivak's work these days, so I am going to suggest a definition from his text, see section 2.5.2.3, which I cannot reproduce here. He uses fiber products.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?

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 $f: S_A \rightarrow S_B$.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?

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 $f: S_A \rightarrow S_B$. I am not sure how to define span composition. I am thinking a lot about David Spivak's work these days, so I am going to suggest a definition from his text, see section 2.5.2.3, which I cannot reproduce here. He uses fiber products.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?

Source Link
Ben Sprott
Ben Sprott
  • 1.3k
  • 14
  • 23

The category of Multisets and Spans: morphism composition and tensor product

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 $f: S_A \rightarrow S_B$.

I have two questions. Firstly, does my definition of the objects and morphisms define a category? Secondly, is it the case that the definition of morphism, as given, also constitutes a tensor product between multisets in the category as defined?