## June 13, 2019

### What’s a One-Object Sesquicategory?

#### Posted by John Baez

A sesquicategory, or $1\frac{1}{2}$-category, is like a 2-category, but without the interchange law relating vertical and horizontal composition of 2-morphisms:

$(\alpha \cdot \beta)(\gamma \cdot \delta) = (\alpha \gamma) \cdot (\beta \delta)$

Better, sesquicategories are categories enriched over $(Cat,\square)$: the category of categories with its “white” tensor product. In the cartesian product of categories $C$ and $D$, namely $C \times D$, we have the law

(1)$(f \times 1)(1 \times g) = (1 \times g)(f \times 1)$

and we can define $f \times g$ to be either of these. In the white tensor product $C \square D$ we do not have this law, and $f \times g$ makes no sense.

What’s a one-object sesquicategory?

A one-object sesquicategory is like a strict monoidal category, but without the law

$(f \otimes 1)(1 \otimes g) = (1 \otimes g)(f \otimes 1)$

I seem to have run into a bunch of interesting examples. Is there some name for these gadgets?

If not, I may take the “one-and-a-half” joke embedded in the word “sesquicategory”, and subtract one. That would make these things semi-monoidal categories.

(The name “white” tensor product is part of another string of jokes, involving the white, Gray, and black tensor products of 2-categories. The white tensor product is also called the “funny” tensor product.)

Posted at June 13, 2019 6:29 PM UTC

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### Re: What’s a One-Object Sesquicategory?

Is there some name for these gadgets?

Yes! These (in the not-necessarily-strict version) were named premonoidal categories by John Power and collaborators. Now I’m curious what kind of examples you have run into.

Posted by: Noam Zeilberger on June 13, 2019 7:08 PM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Cool! Thanks very much. I assume this is a typo in the nLab:

A strict premonoidal category is a monoidal category such that…

They must mean something like

A strict premonoidal category is a premonoidal category such that…

I’ll present my examples here once I get some stuff straightened out.

Posted by: John Baez on June 13, 2019 8:29 PM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Okay, here’s where we ran into a bunch of premonoidal categories:

They show up in Section 4.

Posted by: John Baez on June 16, 2019 1:12 AM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Nice. It would be interesting to see how this fits into the known ways of building premonoidal categories.

In a symmetric monoidal category $C$, if we take an object $s$, then we can form a premonoidal category $C_s$ with the same objects as $C$ but where a morphism $a\to b$ in $C_s$ is a morphism $s\otimes a\to s\otimes b$ in $C$. This is often called (linear) state passing.

More generally, if a symmetric monoidal category $V$ acts on a category $C$, and $s$ is in $C$, then we can form a premonoidal category $V_s$ with the same objects as $V$ but where a morphism $a\to b$ in $V_s$ is a morphism $s\bullet a\to s\bullet b$ in $C$. This is sometimes called linear-use state passing.

(Hope I’ve got this right.) This seems similar to your construction, but formally different, but is it different? Sorry, it’s taking me a while to unpack your definitions.

Posted by: Sam Staton on June 16, 2019 8:37 AM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

We are doing a special case of what you’re calling ‘linear state passing’, so I’d greatly appreciate any references on that.

We’re doing in it in a special case where the symmetric monoidal category $C$ is the free commutative monoidal category on a Petri net, and the object you’re calling $s$ is a collection of ‘catalyst’ tokens.

Posted by: John Baez on June 16, 2019 9:48 PM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Hi, linear state passing is Example 3.4 of Power and Robinson (ps). Many other people have studied it (including myself). Alan Jeffrey came up with a string diagram notation for premonoidal categories based on it. That might fit your “catalyst” intuition.

State passing is at least as old as Scott and Strachey, e.g. The varieties of programming language. Strachey already noticed that using cartesian products isn’t quite right – you shouldn’t really be able to duplicate or discard the “state”.

Posted by: Sam Staton on June 17, 2019 9:55 PM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

What you’re saying reminds me of linear logic. Is the “linear” the same “linear”? Valeria de Paiva has used Petri nets to give a model of linear logic. I wonder if there’s a connection.

Posted by: Joe Moeller on June 19, 2019 1:40 AM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Yes, the same “linear”. (Although “linear logic” was about 10 years after Strachey, as far as I understand.)

Posted by: Sam Staton on June 19, 2019 9:19 AM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

It’s interesting to note that Power and Robinson considered their Example 3.4 to be a more general reformulation of Moggi’s side-effects monad, where the base category C need not be closed.

Posted by: LMR on June 20, 2019 5:38 AM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Thanks, Sam! If I may bother you yet again, who was the first to write something that calls the trick in Example 3.4 of Power and Robinson’s paper ‘linear state passing’? It seems they don’t use this term themselves. I’ll definitely cite this:

• R. E. Møgelberg and S. Staton, Linear usage of state, Logical Methods in Computer Science 10 (2014), lmcs:743. Also available as arXiv:1403.1477.

which you say is an expanded version of something else, but were you the first to make the connection between premonoidal categories and linear state passing explicit?

Posted by: John Baez on June 20, 2019 9:02 PM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

I’m not sure! As LMR hinted, when a monoidal category is closed, there’s a connection to the Kleisli category of the monad $(s\otimes (-))^s$, which is generally called the state monad when the structure is cartesian, and so it’s natural to call it a “linear state monad” in the monoidal case. For example, Masahito Hasegawa, FLOPS 2002. I’m sure that’s why Power and Robinson used the letter $S$. I think O’Hearn and Reynolds were early to point out the importance of linear logic and state (e.g. JACM 2000), although I can’t find “linear state passing” explicitly in that paper. Sorry if that’s not very helpful.

By the way, I think it’s often helpful to look at Freyd categories instead of premonoidal categories. A Freyd category is a premonoidal category with a chosen centre. To be precise a Freyd category is given by (1) a monoidal category $V$, (2) a premonoidal category $C$, (3) an identity-on-objects premonoidal functor $V\to C$. One can also give an equivalent definition without mentioning the funny tensor product: a Freyd category is (1) a monoidal category $V$, (2) a category $C$ with an action of $V$, (3) an action preserving functor $V\to C$. I don’t know whether this is useful to you, nor whether this comes up in sesquicategories.

Posted by: Sam Staton on June 21, 2019 7:12 AM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Thanks! I’ve updated our paper to incorporate some of the things you’ve told me, and acknowledged you. The concept of center seems a bit surplus to requirements for what we’re doing.

We added a couple of questions about premonoidal categories at the end.

Posted by: John Baez on June 22, 2019 2:15 AM | Permalink | Reply to this

### Re: What’s a One-Object Sesquicategory?

Re. your question on a string diagram calculus for premonoidal categories: This isn’t exactly what you’re looking for, but Google turned up a fairly new paper by Lu et al. (Applied Categorical Structures paper here, arXiv version here) on a graphical calculus for “semi-groupal” categories, or semi-group objects in Cat.

Unfortunately, they did prove that the interchange law hold, but it seems to me that they’re working in a more general setting, as they only have an associative tensor product and no units.

Posted by: LMR on June 25, 2019 7:55 AM | Permalink | Reply to this

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