Monoids

Prerequisites: Group Theory. Full path (2 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

Why is it interesting?

Monoids abound in mathematics; they are in a sense the most primitive interesting algebraic structures.http://math.ucr.edu/home/baez/week74.html

Intuitive

A monoid describes processes that can be done one after the other, where doing nothing counts as a process, but where a process cannot necessarily be undone.

Take the operations you can apply to a sheet of paper: rotate it, fold it, cut it. Doing two operations in a row is again an operation, and grouping does not matter: (fold, then cut) followed by a rotation is the same as a fold followed by (cut, then rotate). Leaving the paper alone is the "do nothing" operation. But a cut cannot be reversed. If every operation could be reversed, we would have a group. The monoid is what remains when we drop that requirement.

Simple examples are the counting numbers zero, one, two and so on with addition (adding zero does nothing, but once you have added three there is no counting number you can add to get back to where you started), and words in an alphabet, where the operation is writing one word after another and the empty word does nothing.

In physics, monoids appear whenever information is lost along the way. Diffusion of heat smooths out a temperature profile. Running the heat flow forward for one minute and then for two more minutes is the same as running it for three minutes, but there is in general no way to run it backwards and recover a sharp initial profile. The time evolutions of such systems form a monoid (in the physics literature usually called a semigroup). The renormalization group is another example: coarse-graining throws away short-distance details, so the "group" of coarse-graining steps has no inverses.

Concrete

A monoid is a generalisation of group, in that the requirement that each element has an inverse is dropped.

what is a category with one object? It is a "monoid". The usual definition of a monoid is this: a set M with an associative binary product and a unit element 1 such that a1 = 1a = a for all a in S. […]

We tend to think of this ability to "undo" any process as a key aspect of symmetry. I.e., if we rotate a beer bottle, we can rotate it back so it was just as it was before. We don't tend to think of SMASHING the beer bottle as a symmetry, because it can't be undone. But while processes that can be undone are especially interesting, it's also nice to consider other ones… so for a full understanding of symmetry we should really study monoids as well as groups.

But we also should be interested in "partially defined" processes, processes that can be done only if the initial conditions are right. This is where categories come in! Suppose that we have a bunch of boxes, and a bunch of processes we can do to a bottle in one box to turn it into a bottle in another box: for example, "take the bottle out of box x, rotate it 90 degrees clockwise, and put it in box y". We can then think of the boxes as objects and the processes as morphisms: a process that turns a bottle in box x to a bottle in box y is a morphism f: x → y. We can only do a morphism f: x → y to a bottle in box x, not to a bottle in any other box, so f is a "partially defined" process. This implies we can only compose f: x → y and g: u → v to get fg: x → v if y = u.

[A] monoid is like a group, but the "symmetries" no longer need be invertible; a category is like a monoid, but the "symmetries" no longer need to be composable!

Note for physicists: the operation of "evolving initial data from one spacelike slice to another" is a good example of a "partially defined" process: it only applies to initial data on that particular spacelike slice. So dynamics in special relativity is most naturally described using groupoids. Only after pretending that all the spacelike slices are the same can we pretend we are using a group. It is very common to pretend that groupoids are groups, since groups are more familiar, but often insight is lost in the process. Also, one can only pretend a groupoid is a group if all its objects are isomorphic. Groupoids really are more general.

http://math.ucr.edu/home/baez/week74.html

Abstract

A monoid is a set $M$ with an associative multiplication $M \times M \to M$ and a unit $e$ with $e m = m e = m$. Equivalently, it is a category with a single object $\ast$, where $M = \mathrm{Hom}(\ast, \ast)$ and multiplication is composition. A monoid homomorphism is a functor between such categories. An action of $M$ on a set $X$ is a functor from the one-object category to $\mathbf{Set}$.

The definition makes sense in any monoidal category $(\mathcal{C}, \otimes, I)$. A monoid object is an object $M$ with morphisms

$$\mu : M \otimes M \to M, \qquad \eta : I \to M ,$$

such that $\mu \circ (\mu \otimes 1_M) = \mu \circ (1_M \otimes \mu)$ (after the associator) and $\mu \circ (\eta \otimes 1_M) = 1_M = \mu \circ (1_M \otimes \eta)$ (after the unitors). Different ambient categories give familiar structures:

• In $(\mathbf{Set}, \times, \{\ast\})$: ordinary monoids.
• In $(\mathbf{Ab}, \otimes_{\mathbb{Z}}, \mathbb{Z})$: rings with unit.
• In $(\mathbf{Vect}_k, \otimes, k)$: unital associative algebras over $k$.
• In the category of endofunctors of a category $\mathcal{C}$, with composition as the monoidal product: monads on $\mathcal{C}$.

Free monoids and group completion. The forgetful functor from monoids to sets has a left adjoint: the free monoid on a set $A$ is the set $A^\ast$ of finite words in $A$ with concatenation. In the other direction, every commutative monoid $M$ has a universal map to an abelian group, its Grothendieck group $K(M)$, built from formal differences. The integers are $K(\mathbb{N})$, and the topological K-theory $K^0(X)$ of a compact space $X$ is the Grothendieck group of the monoid of isomorphism classes of vector bundles over $X$ under direct sum.

One-parameter semigroups. In analysis and physics the central example is a strongly continuous family of bounded operators $T(t)$, $t \ge 0$, on a Banach space with

$$T(0) = 1, \qquad T(t + s) = T(t)\, T(s) ,$$

that is, a monoid homomorphism from $(\mathbb{R}_{\ge 0}, +)$ to the bounded operators. It has a generator $A$, defined as $A x = \lim_{t \to 0^+} (T(t) x - x)/t$ on the subspace where the limit exists, and formally $T(t) = e^{tA}$. The Hille-Yosida theorem characterizes which closed, densely defined operators generate contraction semigroups. Examples are the heat semigroup $e^{t \Delta}$ and the dynamics of open quantum systems. When $T(t)$ extends to all $t \in \mathbb{R}$ with $T(-t) = T(t)^{-1}$, the monoid is a group, as for unitary time evolution $e^{-iHt/\hbar}$ in closed quantum systems.

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