Time in Mathematics

I claim that it is commonly believed that Mathematical objects can be seen as genuinely static, with no "Platonic" time in which they do genuinely evolve.

Nevertheless time has its place in mathematics:

1. An endomorphism of a set (seen as a set of states of a system) into itself can be seen as evolution of the system in discrete time steps.

2. For a function of a totally ordered set into a set (seen as above) the ordered set can be seen as "time".

3. as the time-like component in Minkowski space

Questions (slightly modified after Qiaochu's comment and Vhailor's answer):

Which other constructs do give you a "time feeling" or give rise to "dynamic intuition" admit a comparable straight-forward interpretation as "time"?

The examples above are set-theoretical ("concrete"). Is there a more abstract modelling of "time", maybe in category theory?

Solution 1:

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.

Solution 2:

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.)