### The Arrow of Time in Cat

#### Posted by David Corfield

Back here we were talking about the symmetry-breaking that takes place in mathematics by the choice of working in $Set$, which John attributed to nothing less than the ‘arrow of time’.

Why do many-to-one but not one-to-many relations get singled out for single treatment and dubbed ‘functions’? Because functions are supposed to be ‘deterministic’: the cause must determine the effect. We don’t care if the effect fails to determine the cause.

Now what is there to be said about the 2-category Cat and its three duals: $Cat^{op}$, $Cat^{co}$ and $Cat^{co op}$?

We can tell a story where coalgebra (in the general sense) was slow to take off because many algebraic structures defined on our favourite $Set$ are boring when the arrows are reversed. For example, each set supports precisely one comonoid structure. So we have to leave $Set$ behind along with any other cartesian monoidal category if we want interesting comonoids, and look at categories such as Vect.

[Question for experts: Is this right? A comonoid in $C$ is a monoid in $C^{op}$ is a monad in $B C^{op}$ is a monad in $(B C)^{co}$ is a comonad in $B C$.]

But slowly people cottoned onto the idea that there’s plenty to say about coalgebras in $Set$ if we take endofunctors with less of an ‘algebraic’ flavour. For example,

- $X \to F(X) = D(X)$, the set of probability distributions on $X$: Markov chain on $X$.
- $X \to F(X) = P(X)$, the powerset on $X$: Binary relation on $X$.
- $X \to F(X) = X^A \times B$: Deterministic automaton.
- $X \to F(X) = P(X^A \times B)$: Nondeterministic automaton.
- $X \to F(X) = A \times X \times X$, for a set of labels $A$: labelled binary trees.

Now then, will we see the same story played out a level higher with $Cat$? Where there are nice juicy examples of 2-algebras (and its variants) from 2-functors which are 2-monads, is it that the dual scene, or 3 dual scenes, are much more barren? Perhaps first something 2-coalgebra-like will emerge in a less cartesian setting. And then yet later people will come to realise that there were plenty of interesting 2-functors on $Cat$ and so 2-coalgebras there all along.

## Re: The Arrow of Time in Cat

Here’s a couple of other things to think about:

OneDiscussions of the 2-place relation “$x \mathop{determines} y$” are frequently confounded by several different senses of the word “determine”.

There are at least these two senses:

What exactly do we mean by “the” cause and “the” effect, anyway?

Do we mean a totality of events that we call “the cause” of a given effect?

Do we mean a totality of events that we call “the effect” of a given cause?

TwoIn the propositions as types analogy, we have a suggestive relation between the function arrow “$\to$” and the implication arrow “$\Rightarrow$”.

But if we say that function arrows are arrows of time, do we really want to say that implication arrows are arrows of time, and in the same sense, too?