I already feel that I can better understand mathematical structures by using diagrams from category theory. The books I am reading only give me the basics and I am interested in learning how to draw less limited diagrams.
I understand these properties: Objects are dots represented by $(\cdot)$, morphisms are arrows represented by $(\rightarrow)$, and natural transformations are "arrows between arrows", represented by $(\Rightarrow)$. I can represent membership of $x \in X$ by an arrow $\underline{1} \xrightarrow{x} X$ where $\underline{1}$ is the singleton set. To show paths commute, I can give paths the same coloring.
Where I feel dissatisfied is the drawing of more intricate properties and diagrams. For instance, I haven't found a satisfactory diagram in the texts I have been reading to illustrate the property of a subset (although I know it involves some arrow to or from the set $\underline{2}$). Here are three toy examples of the sorts of constructions I would like to make into a diagram, but don't know how to do "rigorously". The third and last toy example is the one I would like the most to see a diagram for, so if you are short on time then you can skip the other two.
(1)
"Let $(X, \tau)$ and $(Y, \tau_1)$ be topological spaces and $f$ a function from $X$ into $Y$. Then $f : (X, \tau) \rightarrow (Y, \tau_1)$ is said to be a continuous mapping if for each $U \in \tau_1$, $f^{-1} (U) \in \tau$".
I would like to turn important theorems like this into a neat diagram. Theorems like this are (in my opinion) sometimes awkward and difficult to understand when written in plain English, and even "psuedo-diagrams" that I have drawn have made these theorems easier to understand. For the sake of conversation, let's say that I want to represent some $\tau$ as being both a subset of $X$ and an element of $\tau$ within the same diagram.
(2)
Here's another toy property that I would like to learn how to diagram:
"Let the change of basepoint map be written as:
$\beta_h : \pi_1 (X, x_1) \rightarrow \pi_1(X, x_0) $ such that $\beta_h [f] = [h \cdot f \cdot h^{-1}]$"
Obviously, I could choose the category $\mathrm{Grp}$ and take both of these fundamental groups as being objects and with $\beta_h$ as a singular arrow which represents a homomorphism. However, I would like to "unpack" this construction a little more. To be more specific I could perhaps remove the "index" $h$ in $\beta_h$ and write the left-hand object as being the product $\pi_1(X, x_1) \times I$. Then I could write projection arrows down into each element of the product, and trivially write the $h$ as an arrow from the singleton into $I$. If possible though, I'd rather like a way to directly write the index $h$ as being a "member" somehow of the arrow $\xrightarrow{\beta}$, perhaps by using some kind of natural transformation? On an even more extreme level, is there some intuitive way to "unpack" a diagram in $\mathrm{Grp}$ into a diagram in $\mathrm{Set}$ by turning each group object into a set-based category? How would I draw the functors between each set category then?
(3)
Let's look at one more example:
"(Baire Category Theorem) Let $(X, d)$ be a complete metric space. If $X_1, X_2, ... , X_n , ...$ is a sequence of open dense subsets of $X$, then the set $\bigcap^{\infty}_{n=1} X_n$ is also dense in $X$."
This one I find the most interesting, because there are several elements I struggle to think of a way to create as a diagram. Not only do I have to write $X_\alpha$ as some subset or element of the topological space, but I also have to take an arbitrary quantity of them. Then I have to perform the intersection operation, which I tentatively think might be an arrow from some product $\alpha \times \alpha \times \cdots \times \alpha$ into some other object. Finally, if possible I would like to either represent density as a structure on the topological space that is connected to my diagram, or somehow condense it as an object or morphism in my diagram.
(To Wit)
I would like to learn how to create more elaborate diagrams to illustrate more complex and interesting structure within mathematics. I want instructions on how to draw some of the diagrams above, or good sources that are very diagram-heavy and show me how to create similar constructions to these. I would also be interested in any fields of study or sources that use "proof by diagram" instead of writing tautological statements in plain English that can be "compiled" into first order logic, if that even exists.