Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > nLab General Discussions: organization of distributivity

Bottom of Page

1 to 11 of 11

- CommentRowNumber1.
- CommentAuthorMike Shulman
- CommentTimeJul 15th 2013
- PermaLink
Author: Mike Shulman
Format: MarkdownItexThere are a lot of different ways in which a category with two monoidal structures can be "distributive". As existing entries we have: * [[distributive category]], where cartesian products distribute over coproducts up to isomorphism. * [[rig category]] and [[bimonoidal category]] --- which appear to be duplicates unaware of each other --- in which one abstract monoidal structure distributes over another up to isomorphism. * [[bipermutative category]], a semistrict version of the previous (although the "examples" given on that page are *not* semistrict). * [[linearly distributive category]], in which the form of the distributive law is "linearized" and it holds only laxly. * One might argue that a [[duoidal category]] or [[2-fold monoidal category]] has a sort of "distributivity", I guess. Two others which we don't have pages for (I don't think), but which are important: * A monoidal category with coproducts whose tensor product preserves coproducts in each variable. Does this even have a standard name? $\otimes$-distributive? * A "colax distributive category", which is like a rig category only the distributivity maps are not assumed invertible. At CT13 we learned that these are another good context in which to talk about near-rings. When I get a chance, I will probably merge [[bimonoidal category]] into [[rig category]], do some inter-linking, and perhaps create pages for the last two. However, it would also be nice to have a list of all these kinds of category in one place and how they compare to each other. I'm inclined to put this on an "index" page of its own, analogous to say the page [[additive and abelian categories]] --- thoughts?
There are a lot of different ways in which a category with two monoidal structures can be “distributive”. As existing entries we have:
- distributive category, where cartesian products distribute over coproducts up to isomorphism.
- rig category and bimonoidal category — which appear to be duplicates unaware of each other — in which one abstract monoidal structure distributes over another up to isomorphism.
- bipermutative category, a semistrict version of the previous (although the “examples” given on that page are not semistrict).
- linearly distributive category, in which the form of the distributive law is “linearized” and it holds only laxly.
- One might argue that a duoidal category or 2-fold monoidal category has a sort of “distributivity”, I guess.
Two others which we don’t have pages for (I don’t think), but which are important:
- A monoidal category with coproducts whose tensor product preserves coproducts in each variable. Does this even have a standard name? $\otimes$ -distributive?
- A “colax distributive category”, which is like a rig category only the distributivity maps are not assumed invertible. At CT13 we learned that these are another good context in which to talk about near-rings.
When I get a chance, I will probably merge bimonoidal category into rig category, do some inter-linking, and perhaps create pages for the last two. However, it would also be nice to have a list of all these kinds of category in one place and how they compare to each other. I’m inclined to put this on an “index” page of its own, analogous to say the page additive and abelian categories — thoughts?
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeJul 15th 2013
- (edited Jul 15th 2013)
- PermaLink
Author: Urs
Format: MarkdownItexSounds good! (I have edited the Examples-section at [[bipermutative category]], adding two genuine examples. Of course more can and should eventually be said here.)

Sounds good!

(I have edited the Examples-section at bipermutative category, adding two genuine examples. Of course more can and should eventually be said here.)
- CommentRowNumber3.
- CommentAuthorMike Shulman
- CommentTimeJul 15th 2013
- PermaLink
Author: Mike Shulman
Format: MarkdownItexOk, created [[distributivity for monoidal structures]] and merged [[bimonoidal category]] into [[rig category]]. Still a couple of gray links to fill in.

Ok, created distributivity for monoidal structures and merged bimonoidal category into rig category. Still a couple of gray links to fill in.
- CommentRowNumber4.
- CommentAuthorMike Shulman
- CommentTimeJul 17th 2013
- PermaLink
Author: Mike Shulman
Format: MarkdownItexApparently I totally forgot about the discussion [here](http://nforum.mathforge.org/discussion/1407/bimonoidal-category/?Focus=34497#Comment_34497). At least the two pages finally got merged.

Apparently I totally forgot about the discussion here. At least the two pages finally got merged.
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeJul 17th 2013
- (edited Jul 17th 2013)
- PermaLink
Author: Urs
Format: MarkdownItexSome little editing I did at _[[rig category]]_: * removed the duplication of the references to Laplaza and Kelly, added anchors in the References-section and links to them instead; * corrected the +-- {: .un_thm} to +-- {: .un_theorem} in order for the formatting to come out right A question: is "Baez's conjecture" not rather a desirable requirement on the definitions? It would be a conjecture if it stated an actual definition instead of saying what a definition should achieve. Related to this, at _[[2-ring]]_ where the article * Alexandru Chirvasitu, Theo Johnson-Freyd, _The fundamental pro-groupoid of an affine 2-scheme_ ([arXiv:1105.3104](http://arxiv.org/abs/1105.3104)) is reviewed, example 4 asserts this "conjecture" as example 2.3.4 in this article, for their definitions (which look rather reasonable).
Some little editing I did at rig category:
- removed the duplication of the references to Laplaza and Kelly, added anchors in the References-section and links to them instead;
- corrected the
```
  +-- {: .un_thm}
```
  to
```
  +-- {: .un_theorem}
```
  in order for the formatting to come out right
A question: is “Baez’s conjecture” not rather a desirable requirement on the definitions? It would be a conjecture if it stated an actual definition instead of saying what a definition should achieve.

Related to this, at 2-ring where the article
- Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme (arXiv:1105.3104)
is reviewed, example 4 asserts this “conjecture” as example 2.3.4 in this article, for their definitions (which look rather reasonable).
- CommentRowNumber6.
- CommentAuthorMike Shulman
- CommentTimeJul 17th 2013
- PermaLink
Author: Mike Shulman
Format: MarkdownItexI know nothing about "Baez's conjecture" -- it was apparently inserted by John himself in the first version of the page, three years ago. However, the 2-category of symmetric rig categories is essentially determined by the definition of symmetric rig category; it's not something we get to choose in order to make a conjecture come out true.

I know nothing about “Baez’s conjecture” – it was apparently inserted by John himself in the first version of the page, three years ago. However, the 2-category of symmetric rig categories is essentially determined by the definition of symmetric rig category; it’s not something we get to choose in order to make a conjecture come out true.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeJul 17th 2013
- (edited Jul 17th 2013)
- PermaLink
Author: Urs
Format: MarkdownItexOkay. I'd be inclined to replace it by a statement that says something like the following: > The analog role in 2-rigs to the role played by the natural numbers among ordinary rigs should be played by the standard categorification of the natural numbers: the category of finite sets, or at least its core. One is inclined to demand that a reasonable definition of 2-rigs should be such that $Core(FinSet)$ is the initial object in that 2-category. For some definitions this has been show (and here goes a pointer to Johnson-Freyd.) What do you think?

Okay. I’d be inclined to replace it by a statement that says something like the following:

The analog role in 2-rigs to the role played by the natural numbers among ordinary rigs should be played by the standard categorification of the natural numbers: the category of finite sets, or at least its core. One is inclined to demand that a reasonable definition of 2-rigs should be such that $Core(FinSet)$ is the initial object in that 2-category. For some definitions this has been show (and here goes a pointer to Johnson-Freyd.)

What do you think?
- CommentRowNumber8.
- CommentAuthorMike Shulman
- CommentTimeJul 18th 2013
- PermaLink
Author: Mike Shulman
Format: MarkdownItexSure. (Do Chirvasitu and Johnson-Freyd actually show it, or merely assert it?)

Sure. (Do Chirvasitu and Johnson-Freyd actually show it, or merely assert it?)
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeJul 18th 2013
- (edited Jul 18th 2013)
- PermaLink
Author: Urs
Format: MarkdownItexOkay, check out what I did: I moved it to _[Propoerties -- Initial object](http://ncatlab.org/nlab/show/2-rig#PropertiesInitialObject)_. Feel invited to modify. > Do Chirvasitu and Johnson-Freyd actually show it, or merely assert it? Yeah, they just assert it, I think. They seem to think it's obvious. In the above I have now put it like this: > For the notion in def. 1 this was conjectured by John Baez, for the notion in def. 3 this is asserted in (Chirvasitu & Johnson-Freyd, example 2.3.4).

Okay, check out what I did: I moved it to Propoerties – Initial object. Feel invited to modify.

Do Chirvasitu and Johnson-Freyd actually show it, or merely assert it?

Yeah, they just assert it, I think. They seem to think it’s obvious. In the above I have now put it like this:

For the notion in def. 1 this was conjectured by John Baez, for the notion in def. 3 this is asserted in (Chirvasitu & Johnson-Freyd, example 2.3.4).
- CommentRowNumber10.
- CommentAuthorMike Shulman
- CommentTimeJul 18th 2013
- PermaLink
Author: Mike Shulman
Format: MarkdownItexWait, now something's not right: $core(FinSet)$ is not cocomplete, so it can't be a 2-rig in the sense of either of those definitions. Aren't both the conjecture and the assertion about $Set$ itself?

Wait, now something’s not right: $core(FinSet)$ is not cocomplete, so it can’t be a 2-rig in the sense of either of those definitions. Aren’t both the conjecture and the assertion about $Set$ itself?
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeJul 18th 2013
- PermaLink
Author: Urs
Format: MarkdownItexSure, I wasn't thinking. Fixed now.

Sure, I wasn’t thinking. Fixed now.

1 to 11 of 11

nForum

Discussion Feed

Not signed in

Site Tag Cloud

nLab > nLab General Discussions: organization of distributivity