Add total/cototal category properties (WIP)

[dschepler]

All categories decided for "total" property.

Unknown categories for "cototal" property:
category of commutative monoids
category of Hausdorff spaces
category of locally ringed spaces
category of semigroups

[dschepler]

I have some rough ideas on some of the others: on Hausdorff spaces and semigroups, I think I should be able to use an idea similar to the one for Cat to keep control over the images of constant maps. For CMon, I think the "subdirectly irreducible" property might have to do with limiting the number of maps to it - though I'm not yet at all sure how to translate that into a contradiction. And on locally ringed spaces, I have a vague idea that I might be able to define a functor whose L(T) would have a number of maps from Spec k which grows faster than possible for any single locally ringed space.

Anyway, no rush on reviewing this - I was just working on this off and on over the past week, and wanted to get the progress so far pushed before resuming work on the quasitopos PR.

[ScriptRaccoon]

cf. #259, this property now requires at least one tag (listed in config.yaml under category_property_tags). If no existing tags fit, create a new one.

Same with cototal.

[dschepler]

OK, off the top of my head, totality is closely related to certain non-small colimits and limits existing. Namely, say a diagram $X : \mathcal{I} \to \mathcal{C}$ (with $\mathcal{I}$ not necessarily small) "has residually small coslices" if for each object $c$ of $\mathcal{C}$, the collection of connected components of $c \downarrow X$ is bijective to a set. Then a category is total if and only if every diagram with residually small coslices has a colimit. And it also has the consequence (but probably not an equivalence) that if $X : \mathcal{I} \to \mathcal{C}$ has the property that the collection of cones $c \to X_i$ is bijective to a small set for each object $c$ of $\mathcal{C}$, then the diagram $X$ has a limit.

Similarly, since the Yoneda embedding always preserves limits, the totality property (at least if you already know the category is complete and cocomplete) is purely about size issues, so something similar to the solution set condition in the Freyd adjoint functor theorem.

So, I'm thinking tags along the lines of "limits", "colimits" and "size".