What are examples of rig categories which

- are not just rigs or ordered rigs, and
- are not distributive monoidal categories (so the sum is not the coproduct)?

I cannot seem to come up with an example myself except very artificial ones (such as taking a distributive monoidal category and removing some arrows). I’m sure there’s quite some though, can anyone help?

]]>In the discussion at the bottom of monoidal category, we read:

In fact a strict monoidal category is just a monoid internal to the category Cat. Unfortunately this definition is circular, since to define a monoid internal to Cat, we need to use the fact that Cat is a monoidal category!

And then later

For example, you can define a monoidal category to be a pseudomonoid internal to the 2-category Cat — but nobody knew how to define these concepts until they knew what a monoidal category is!

Doesn’t the same circularity afflict the definition of monoidal category that’s on the page? For example, the associator is given as

$a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))$But this doesn’t make sense if taken literally. You cannot have a natural transformation between functors on different domains, and the domain of these functors are not the same. The domain of the left functor is $(\mathcal{C}\times \mathcal{C})\times\mathcal{C}$ whereas the domain of the right functor is $\mathcal{C}\times (\mathcal{C}\times\mathcal{C})$. Of course those two categories are isomorphic, and using that isomorphism, we can make sense of the definition. But that’s using the monoidal structure of $\text{Cat}$! We’re being circular in exactly the same way as we would if we defined a monoidal category as a (pseudo)monoid in the monoidal (2-)category $\text{Cat}$!

I guess it’s not circular in any formal sense, since we can just observe that any cartesian category has canonical isomorphisms $(\mathcal{C}\times \mathcal{C})\times\mathcal{C}\cong \mathcal{C}\times (\mathcal{C}\times\mathcal{C})$ and we can just insert that into the definition as needed, without commenting that it is part of a monoidal structure on the ambient category. The same applies to any monoid in any cartesian category. In particular, I think it’s not formally circular to define a (weak/strict) monoidal category as a (pseudo) monoid in $\text{Cat}$.

Shouldn’t that equivalent definition be mentioned higher in the article, since it’s valid and not really circular?

And shouldn’t the article be more explicit about this, about using the cartesian associator and unitors of $\text{Cat}$, given that it’s basically an article about the need to be careful and rigorous about the axioms of associators and monoidal structures?

Also, is there some more coherent, higher categorical way out of this circularity, other than than just capping it with a cartesian structure at some level of the higher categorical ladder?

]]>I could not understand the proof of the fact that if a vector space $V$ over a field $k$ is dualisable in the monoidal category $\mathbf{Vect}_k$ then it should be finite dimensional. I understand that the image of $k$ under the unit is a $1$-dimensional subspace, but why should this imply the finite dimensionality of $V$? I would be thankful if someone could provide me the complete proof or a pointer to it.

With my regards,

partha

]]>http://math.stackexchange.com/questions/1992156/how-to-express-predicate-logic-in-the-categorical-monoidal-logics

Maybe someone knows the asnwer?

The question is:

Rosetta stone in the book "New Structures in Physics" (http://www.springer.com/la/book/9783642128202) is the correspondence between the propositional (linear) logic from the one side and the monoidal categories from the other side. Each proposition can be represented as the object, each deduction (proof) among two literals can be represented as the morphism between repsective objects and each connective can be attributed to the tensor product among objects in category. That works for propositional logic.

The similar correspondence can be extended to the modal logics and modal operators, e.g. in paper http://www.cs.nott.ac.uk/~psznza/papers/Alechina++:01a.pdf

The question is: can this correspondence between logic (deductive system) and the category theory be extended to the predicate logic? There is article where it has been done (it is said so) http://amcm.pcz.pl/get.php?article=2015_1/art_03.pdf This extension is done in chapter 4 of this article. The problem is - I don't understand this chapter from the very basic things. The authors say that predicate can be expresses by the subset of the object - but there is no such notion in the category theory. Maybe someone can explain chapter 4 or give alternative correspondence between predicate logic and category theory?

This correspondence is very exciting subject because it can lead to the universal logic and universal reasoning system - very much welcome in the applied AI! ]]>

I want to attempt to apply my theory of funcoids to study of integral curves of different smoothness classes. For a start I consider curves in $\mathbb{R}^n$ for finite $n$.

Below I will ask a question about staroids. But after this I describe this in terminology of common knowledge (without using funcoids or staroids), for these who has not read my book.

For this I probably need to well understand $n$-staroids for finite $n$ ($2$-staroid is essentially the same as funcoid).

Denote $\mathsf{Strd}(-,-)$ (where the poset $\mathsf{Strd}(S_0, \dots, S_{n-1})$ that is a staroid between posets $S_0, \dots, S_{n-1}$ looks like to be an unbiased tensor product).

I want to prove that $\mathsf{Strd}(-,-)$ is a tensor product in the category $\mathbf{Pos}$.

Something equivalent to this is claimed (for the special case of join-semilattices) in this this MathOverflow answer.

However I have some trouble to write down a proof.

I ask for help to prove that $\mathsf{Strd}(-,-)$ is a tensor product. It should be a consequence of the (to be proved) fact that $\mathsf{Strd}(A,B,C) = \mathsf{Strd}(\mathsf{Strd}(A,B), C)$.

The first (easy) step (for the special case of join-semilattices) is to note that a staroid between join-semilattices $S_0, \dots, S_{n-1}$ of finite arity $n$ is a multi-join-homomorphism (a join-semilattice homomorphism in each argument separately) $\prod_{i\in n} S_i \rightarrow 2$ for join-semilattices $S_i$.

(The last paragraph describes it in a language without funcoids and staroids. So you can answer my question even if you have not read my writings.)

Then the MathOverflow answer states that this is equivalent to order homomorphism $\bigotimes_{i\in n} S_i \rightarrow 2$ (where $\bigotimes$ is a tensor product). I don’t understand why.

Moreover when I tried to prove $\mathsf{Strd}(A,B,C) = \mathsf{Strd}(\mathsf{Strd}(A,B), C)$, I had a trouble leading me to think that it isn’t an equality (and then $\mathsf{Strd}(-,-)$ would be not a tensor product), but I remembered that this is stated in that MO question. I think the error is on my part, not the MO answer, and thus it is indeed a tensor product (and this would be a beautiful result), but I don’t understand what exactly is wrong with my reasoning. I probably need help.

I want an explicit proof (not something about ring or distributive lattices) for the following reasons:

- It may be generalizable for wider case of any posets rather than join-semilattice only.
- I don’t know ring theory well enough.

John Wiltshire-Gordon and I were reading the nlab article on the monoidal Dold-Kan correspondence and we are puzzled by the following statement:

“The two functors in the Dold-Kan correspondence individually respect these monoidal structures, in the sense that they are lax monoidal functors.”

This seems to us to be false. Let $D_1$ be the chain complex (indexed from $0$) which is $\mathbb{Q}$ in degree $1$ and $0$ in every other degree. Then $D_1 \otimes D_1$ (using the monoidal structure on chain complexes, by taking the total complex of the double complex) is the chain complex which is $\mathbb{Q}$ in degree $2$ and $0$ in every other degree. Applying $\Gamma$, we get a certain simplicial vector space which has dimension $\binom{n-1}{2}$ in degree $n$.

On the other hand $\Gamma(D_1)$ is a simplicial vector space which has dimension $n$ in degree $n$ and $\Gamma(D_1) \otimes \Gamma(D_1)$ (using the monomial structure on simplicial vector spaces, by pulling back along the diagonal map) has dimension $n^2$ in degree $n$. So $\Gamma(D_1 \otimes D_1)$ is not isomorphic to $\Gamma(D_1) \otimes \Gamma(D_1)$. Of course, the two are homotopy equivalent.

Are we missing something dumb, or is this an error in the article?

]]>Have added the “definition” of a symmetric monoidal $(\infty,n)$-category to the entry.

]]>Has anyone read Vicary's Categorical Quantum harmonic oscillator? I am attempting a calculation using the adjunction RQ which he develops in that paper. Specifically, I am using the category of comonoids Cx (which he describes in the paper) as if it were SET and I intend to do some set theory in Cx. He already demonstrates Cx has finite products, a 0 and a 1 object. I am looking for the 2 object, but really I want these so that I can have the powerset functor. What I am interested in are unions, intersections and powersets. I have Lawvere and Rosebrugh here and looked up these structures, but if anyone can help out, that would be great. Thanks. ]]>

I’m looking for references on the structure which can be roughtly described as follows: given a (braided? symmetric?) monoidal category $C$, I want to consider a simplicial set $N(\mathbf{B}C)$ with a single vertex, an edge for every object of $C$, a triangle with edges $X,Y,Z$ for every morphism $phi:Z\to X\otimes Y$, a tethraedron for every four triangles making up a commutative diagram involving the associator of $C$, higher coherences..

Any suggestion? thanks

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