Time in Mathematics

I know of two other concepts that have a "time" feeling to them.

The definition for homotopy involves a parameter which can very intuitively be interpreted as time.

A lie group also has to me some feeling of time, since it adds on top of classical geometry the idea that isometries must be a sort of continuous motion through time, not just a "teleportation" between two states.

I wish I had this reference handy, but Atiyah said something to the effect that algebra is about time and geometry is about space. The "processes" in mathematics are, in the broadest sense, the things concerned with time. (This doesn't take into account the idea of "reification", the transformation of a process into an object, which is central to mathematical practice.)

In general, anywhere that you see a morphism there is probably a time-like interpretation.

For example, discrete-time dynamical systems could be viewed as a category with one object (the state space) and a distinguished morphism $f$ representing evolution by one time unit, and composition of morphisms giving the evolution by multiple time units.

The identity morphism gives us the concept of 'no time passing' and the associative property of morphism composition (i.e. $f\circ (g\circ h) = (f\circ g)\circ h$) ensures that 'time' behaves in a familiar way (i.e. advancing by $n$ time units and then $m$ time units is the same as advancing $m$ time units and then $m$ time units). In this framework, reversible dynamical systems are naturally seen as categories in which every morphism is invertible (i.e they are groups).

In computer science, the morphisms in a particular category are seen as abstracting the idea of sequential computation (i.e. perform this operation, then this other one). The generalization to arrows takes this further by also allowing parallel and recursive computations, and there are various interesting theorems to be proved about when a computation can be parallelised, and whent he order of two computations matters.