Categorical Time I always want to apply category theory to structures that involve "time" or "stepping"/"increment"(discrete "time").
I visualize it as a sequence of categories that are somehow connected(generally the objects are the same and only the morphisms will change, not sure if they are functorially related or not in all cases).
In my cases they are discrete systems in the sense that a morphism does not continuously flow, although, I suppose it should be valid for such systems too(analogous to the calculus of limits).
Is there anything in category theory that deals with this topic? I find that that category theory is more relevant to real systems when time seems to be included but I haven't been able to really apply it with time. In fact, one thing I sort of want to do is have Markovian like structures on categories. A set of categories with probabilistic transitions between them, this too is a sort of category with, I guess, the transitions being functors that are selected with a probability(but in a sequence of steps).
It seems to be category theory generally is thought of as a static structure but this is quite limiting to problems that are actually relatively simple statically but change over time. E.g., imagine a cell phone network where the objects are cell phones and the morphisms are connections. The connections may be off or on(a phone call) and also last for any length of time. Each instant in time there is a category/graph of the network. But over time the graph is dynamically changing. We can even do things like calculus over it.
So, in such ways it looks more like graph theory, BUT, of course, these happen to be categorical structures(identities, composition inclusion, etc)... so the goal is to extract some time-dependent structural variation. E.g., we can think of a typical "basic" category as static in time analogous to a constant function, it's derivative is 0. In this case the categorical structure is dynamic and its "derivative" is not constant but shows how the structure is evolving. Ultimately I want to be able to extract some meta-structure from the evolution of the changing categorical structure(and ultimately derive some type of meta^n-structural deductive chains which would, I hope, ultimately be categorical in nature).
 A: I don't know if the following are exactly what you're looking for, since they are about modelling dynamical systems within category theory, rather than having a dynamical system of categories, but they are likely to be useful for inspiration, if nothing else.
There's a classic way to look at discrete-time dynamical systems in category theory. One is just to model a dynamical system as a set $X$ with an endomap, that a function $f\colon X\to X$, which you think of as the update function. This can be seen as a functor $\mathcal{N}\to\mathsf{Set}$, where $\mathcal{N}$ is a monoid (i.e. a category with a single object) whose morphisms are the natural numbers and the composition of two morphisms is given by addition. Natural transformations between such functors give a natural notion of morphism between dynamical systems, and the resulting category (being a presheaf category and hence a topos) has a lot of very nice properties. This is explored in Lawvere and Schanuel's classic book 'Conceptual Mathematics', among other places.
For Markov processes, a nice place to start is Tobias Fritz' recent paper A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. This paper gives a modern framework for thinking about probability in category theory, based on what he calls "Markov categories", which generally behave like categories of sets and stochastic functions (i.e. Markov kernels). Similarly to the above construction discrete-time, time-homogeneous Markov process is a functor $\mathcal{N}\to \mathscr{M}$, where $\mathscr{M}$ is a Markov category. Fritz mentions Markov processes only briefly in his paper, but still I think it's a good place to start in thinking about them.
In both of these cases, if you want to consider non-time-homogeneous systems (i.e. where the update function can change on every time step) you can look at functors from $\mathbf{N}$ instead of $\mathcal{N}$, where $\mathbf{N}$ is the natural numbers seen as a preorder instead of a monoid. (I'm making this notation up - I'm not sure if there are standard symbols for these categories.)
This kind of approach can also be used for continuous-time systems, by replacing the monoid of natural numbers with a monoid of real numbers, although things tend to get significantly more complicated due to the need to consider topology. See this blog post by Jade Master for such a construction.
Since you seem to be looking for a discrete-time dynamical system where the state space consists of categories, it could be that you would be interested in functors $\mathcal{N}\to\mathsf{Cat}$ or $\mathbf{N}\to\mathsf{Cat}$. This seems like it should be related to the Grothendieck construction. I have no intuition for what it would be like, and I also don't immediately know how one would make it into a stochastic process.
Going in a somewhat different direction, there is a fair bit of current research in category theory in something called "open dynamical systems", which may be in discrete or continuous time. The idea here is to define dynamical systems with inputs and outputs (in various different senses), such that they can be composed with each other. If you search YouTube you can find some lectures by David Jaz Myers that give a good overview, and you can also find some papers on this topic by John Baez' group, among others.
A: Time is a physical concept and it's a question of how it is to be modelled. It's quite possible that it's an artifact of our modelling that we get time invariance since causal nets, a model in quantum gravity allow, for modelling of time as open - a relatively recent result.
This to me seems a prime suspect in modelling by category theory: the problem being to say something new that hasn't already  been said by causal set theory.
