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Stream: learning: questions

Topic: When is the Alg(T) cocartesian?

Alexander Gietelink Oldenziel (Dec 16 2020 at 17:02):

GIven a monad $T$ on a symmetric monoidal category $C$ a sufficient (and necessary?) condition for the category of Alg(T) to have a symmetric monoidal product $\otimes_T$ such that the forgetful functor $U: Alg(T) \to C$ preserves the tensor product is for the monad to be commutative/monoidal. See e.g. https://ncatlab.org/nlab/show/tensor+product+of+algebras+over+a+commutative+monad

What conditions do we require on the monad $T$ such that said tensor product is actually a coproduct?

I am thinking of the case where $C=k-Vect$ and $T$ is the free algebra monad ('symmetric algebra'). In this case $Alg(T)=k-Alg$, the tensor is a coproduct (so that the category of affine schemes inherits a product structure!).

John Baez (Dec 16 2020 at 17:18):

Just to warn readers: you are using "free algebra" to mean "free commutative algebra" and $k-Alg$ to mean "commutative algebras over the field $k$".

John Baez (Dec 16 2020 at 17:19):

Unless your algebras over a field are commutative it's not true that their tensor product is their coproduct.

John Baez (Dec 16 2020 at 17:21):

One reason the tensor product of commutative algebras over a field is their coproduct is this: the category of commutative monoid objects in any symmetric monoidal category is cocartesian.

John Baez (Dec 16 2020 at 17:21):

The only answers I can imagine to your question rely on this fact.

JS PL (he/him) (Dec 16 2020 at 22:08):

The answer you're looking for of turning the tensor product of the base category into a coproduct in the Eilenberg-Moore category is called a "comonoidal algebra modality", which is the dual of a "monoidal coalgebra modality" which is used in the categorical semantics of linear logic.

JS PL (he/him) (Dec 16 2020 at 22:09):

The dual of your result is found in these notes, where they explain how to show that tensor product in the base category turns into a product in the coEilenberg-Moore category

JS PL (he/him) (Dec 16 2020 at 22:09):

http://www.cs.man.ac.uk/~schalk/notes/llmodel.pdf
you can also find this result in many papers on categorical semantics of linear logic

JS PL (he/him) (Dec 16 2020 at 22:11):

This paper of mine with Blute, Cockett, and Seely also gives the full definition of monoidal coalgebra modalities and string diagrams:
https://arxiv.org/abs/1806.04804
we also talk about algebra modalities explicitly in this paper:
https://arxiv.org/pdf/1910.05617.pdf
and mention that the tensor product becomes a coproduct in the Eilenberg-Moore category.

JS PL (he/him) (Dec 16 2020 at 22:11):

Essentially, let $C$ be a symmetric monoidal category, with tensor $\otimes$ and unit $k$

JS PL (he/him) (Dec 16 2020 at 22:16):

A comonoidal algebra modality is:

1. A monad $(T, \mu, \eta)$, where $T: C \to C$ is the endofunctor, and $\mu_X: TT(X) \to T(X)$ and $\eta_X: X \to T(X)$ are the monad structure maps.
2. Natural transformations $\nabla_X: T(X) \otimes T(X) \to T(X)$ and $u_X: k \to T(X)$ such that $(T(X), \nabla_X, u_X)$ is a commutative monoid and $\mu_X: (TT(X), \nabla_{T(X)}, u_{T(X)}) \to (T(X), \nabla_X, u_X)$ is a monoid morphism
3. A natural transformation $m_{X,Y}: T(X \otimes Y) \to T(X) \otimes T(Y)$ and $m_k: T(k) \to k$ such that $(T, m, m_k)$ is a symmetric comonoidal (lax) functor
4. $\mu$, $\eta$, $\nabla$, and $u$ are all monoidal transformations
5. $\nabla$ and $u$ are also $T$-algebra maps

JS PL (he/him) (Dec 16 2020 at 22:18):

Then the Eilenberg-Moore category of a comonoidal algebra modality is coCartesian where $\otimes$ from the base category becomes a coproduct in the Eilenberg-Moore category.

JS PL (he/him) (Dec 16 2020 at 22:23):

John Baez said:

The only answers I can imagine to your question rely on this fact.

That's correct! There is a forgetful functor from the Eilenberg-Moore category of an algebra modality (part 1 and 2 above) to the category of commutative monoids. Then parts 3,4,5 allow us to say that the tensor product is in fact a coproduct

I talk about algebra modalities and how to turn their algebras into commutative monoids in this paper:
https://arxiv.org/pdf/1803.02304.pdf

JS PL (he/him) (Dec 16 2020 at 22:26):

If we assume the base category also has coproducts, it also follows from the fact that we have isomorphisms $T(A \oplus B) \cong T(A) \otimes T(B)$ and $T(1) \cong k$, where $\oplus$ is the coproduct and $1$ is the initial object.