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Stream: learning: questions

Topic: Asymmetric monoidal categories

Asad Saeeduddin (May 19 2020 at 20:27):

Anyone know where I can get a big list of monoidal categories that lack a symmetry or braiding? The simpler the examples the better.

Asad Saeeduddin (May 19 2020 at 20:28):

The only one I know of off the top of my head is functor composition

John Baez (May 19 2020 at 20:33):

I think the category of representations of a Hopf algebra is a great one. A "Hopf algebra" is just an algebra (over some field, say) equipped with precisely enough structure to make its category of representations ( = modules) be monoidal.

Oscar Cunningham (May 19 2020 at 21:03):

EDIT: Ignore this comment. I must have misread the question because I'm describing how to get a braided category.

Since a braided monoidal category is just a $3$-category with only one object and only one morphism, and the canonical example of a $3$-category is $2$-$\mathbf{Cat}$, you can get one family of examples by picking any $2$-category and looking at the category of natural transformations from the identity on that $2$-category to itself.

Oscar Cunningham (May 19 2020 at 21:04):

Are there any nontrivial examples of this if we pick the $2$-category $\mathbf{Cat}$?

Soichiro Fujii (May 20 2020 at 00:50):

One can obtain a monoidal category (typically without a braiding) from an endo-hom-category of a bicategory.

That is, if $\mathscr{K}$ is a bicategory and $a$ is an object of $\mathscr{K}$, then $\mathscr{K}(a,a)$ is a monoidal category.

So for instance we have:

• the monoidal category $\mathscr{S}pan(X,X)$ of endo-spans on a set $X$ (or more generally on an object of a category with pullbacks),
• the monoidal category $\mathscr{M}od(R,R)$ of $R$-bimodules over a ring $R$ (though if $R$ is commutative, it has a symmetry),
• the monoidal category $\mathscr{P}rof(\mathcal{C},\mathcal{C})$ of endo-profunctors on a category $\mathcal{C}$, etc.

The monoidal category of endofunctors on a category is also a natural example of this, with $\mathscr{K}=\mathscr{C}at$.

John Baez (May 20 2020 at 06:14):

Nice, Soichiro. It's interesting to look at monoids in all these monoidal categories you list. A monoid in the monoidal category of endofunctors of a category is a monad. A monoid in the monoidal category of endo-spans on a set is a category. And so on.

Jules Hedges (May 20 2020 at 09:55):

Free monoidal categories (on a monoidal signature, ie. a bunch of generating objects and morphisms) are an important class of example, their morphisms are string diagrams without crossings. Categories of diagrams without crossing are sometimes important, eg. in some types of quantum computing where crossings are expensive and you don't want to freely allow them. Free autonomous (non-symmetric compact closed) categories and free autonomous posets are important in linguistics

Amar Hadzihasanovic (May 20 2020 at 10:32):

Another good way of producing non-symmetric monoidal categories is to start with a “small” non-symmetric monoidal category $C$ presented by a few generators and relations, and then extend to the category of presheaves on $C$ by Day convolution.

Interesting examples: augmented simplicial sets with the join, cubical sets with the Gray product...

Pastel Raschke (May 20 2020 at 11:00):

ordinal arithmetic (i believe this is ≈ augmented simplicial sets with join? but can be transfinite obviously)

Joe Moeller (May 20 2020 at 20:47):

Here's an extra cheap class of examples: a non-commutative monoid is a discrete non-braided monidal category.

Joe Moeller (May 20 2020 at 20:49):

You can look at monoid objects internal to concrete categories, and these can be turned into ordinary monoids.

Joe Moeller (May 20 2020 at 20:51):

You can feed these into the thing above, presheaves and Day convolution. I'm not sure what these are exactly. A family of objects indexed by the monoid, and tensor given pointwise by the monoid?

Joe Moeller (May 20 2020 at 20:53):

It's probably more interesting to think about presheaves on a monoid thought of as a one object category. This is the category of actions of the monoid.

Joe Moeller (May 20 2020 at 20:53):

Unfortunately for the purpose of this conversation, if you wanted a monoidal one object category that you could then apply Day convolution to, Eckmann-Hilton says it has to be commutative.

Joe Moeller (May 20 2020 at 20:55):

The monoid objects comment earlier points to an enriched version of all this.