Posted on 2019-11-07
by Oleg Grenrus

Semigroups, semigroupoids... are there more semithings?

Let’s start "from the beginning". Groups is 19th century thing. A *group* is a set with a binary operation , identity element and inverse element for each . With an obvious laws you can imagine relating these.

If we remove *identity* element from a group, "obviously" we get a *semigroup*. Because if there’s *any* element , we can recover the identity element by .

So what’s a semigroup with identity element? For sometimes it were just that, until we started call it *monoid*. Go figure, naming is hard, not only in programming, but also in mathematics.

You are hopefully familiar with a concept of a category. I repeat a definition here:

**Definition** (Awodey 1.1) A *category* consist of the following data

Objects:

Arrows:

For each arrow , there are given objects

called the

*domain*and*codomain*of . We writeto indicate that and .

Given arrows and , that is, with

there is given an arrow

called the

*composite*of and .For each object , there is given an arrow

called the

*identity arrow*of .Associativity:

for all , , .

Unit:

for all .

If you think hard (or read a book), you’ll learn that a single object category is a monoid: category arrows are monoid elements, and the laws work out.

The group analogue is called *groupoid*. In addition to category data, we require that for each arrow there is an inverse arrow , such that and . Or more succinctly: that each arrow is an isomorphism.

But we can also remove stuff: if we remove identity arrows, and unit law we get *semigroupoid*.

**Definition** A *semigroupoid* consist of the following data

Objects:

Arrows:

For each arrow , there are given objects

called the

*domain*and*codomain*of . We writeto indicate that and .

Given arrows and , that is, with

there is given an arrow

called the

*composite*of and .Associativity:

for all , , .

A reader probably ask themselves: are there interesting, not-contrived examples of semigroupoids, which aren’t also categories? There are. If poset (set with partial order) is an example of category, then a set with strict order^{1}, is an example of semigroupoid.

As a concrete example, natural numbers with an unique arrow between and when is a semigroupoid.

There are no identity arrow, as , but associativity works out: if and then . Let’s call this semigroupoid .

Finally, a plot twist. nLab calls semigroupoids semicategories, and don’t even mention a semigroupoid as an alternative name!

Recall a definition of functor.

**Definition** (Awodey 1.2) A *functor*

between categories and is a mapping of objects to objects and arrows to arrows, in such a way that

,

,

.

That is, preserves domains and codomains, identity arrows, and composition. A functor thus gives a sort of "picture" – perhaps distorted – of in .

Functors preserve identities, but semigroupoids don’t have identities to be preserved. We need a weaker concept:

**Definition** A *semifunctor*

between semigroupoids and is a mapping of objects to objects and arrows to arrows, in such a way that

,

.

That is, preserves domains and codomains, and composition. A functor thus gives a sort of "picture" – perhaps distorted – of in .

An identity functor is obviously a semifunctor, also a successor functor is a semifunctor .

In Haskell, that would be silly to define a class for (endo)semifunctors:

```
-- semimap f . semimap g = semimap (f . g)
class Semifunctor f where
semimap :: (a -> b) -> f a -> f b
```

It’s the `Functor`

type-class without an identity law. On the other hand, something like

```
data Mag a b t where
Map :: (x -> t) -> Mag a b x -> Mag a b t
One :: a -> Mag a b b
instance Semifunctor (Mag a b) where
fmap f (Map g x) = Map (f . g) x
fmap f x = Map f x
```

would be valid.

Now as we have semifunctors, would it make sense to ask whether endosemifunctors can form a monad

**Semimonad** A *semimonad* on semigroupoid consists of

an endosemifunctor

semi natural transformation (

`return`

)semi natural transformation (

`join`

)Associativity (as semi natural transformations )

Identity (as semi natural transformations )

Looks a lot like monad, but for semifunctors. I have an example: the semifunctor is a semimonad. Feels like (I didn’t check) that all strictly monotonic functions would fit. We need to find some more structured semigroupoid than to find more interesting semimonads, but I haven’t yet.

I end with a catchy phrase:

*A semimonad is a monoid (!) in the category of endosemifunctors.*

What would be a semigroup in the category of endofunctors^{2}, or a semigroup in the category of endosemifunctors?

Naming is hard.

There’s a paper *The theory of semi-functors* by Raymond Hoofman https://doi.org/10.1017/S096012950000013X.

http://mathworld.wolfram.com/StrictOrder.html, I not dare to call the set with strict order a soset↩

in Haskell we call it

`Bind`

, at least now (or`Apply`

for different semigroup)↩

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