Add accessible categories and related properties

[ScriptRaccoon]

ℹ️ This is a rewrite of #59 by @ykawase5048 with a clean commit history, closer to the main branch and ready to be merged. See there for details what this PR is about and the discussions around it.

I have added "accessble" and several related properties:

• (basic): accessible, ℵ₁-accessible, finitely accessible
• (dual): coaccessible, locally copresentable
• (related to sifted colimits): generalized variety, multialgebraic
• (accessible with some limit): locally multipresentable, locally finitely multipresentable, locally polypresentable
• (multilimits): multilimits, multiterminal object, multicolimits, multiinitial object
• (size): countable, essentially countable

Follow Up: #91

[ykawase5048]

Thanks for rewriting. This looks good!

[ScriptRaccoon]

@ykawase5048 🙏🏻

[ScriptRaccoon]

@ykawase5048 I was just checking if a certain category has multi-colimits and stumbled about a detail, maybe you can help.

At some point in this PR we said that a multi-colimit is a multi-initial object in the category of cocones. This must be a set of cocones, right?

Does a large discrete category have a multi-initial object?

[ykawase5048]

At some point in this PR we said that a multi-colimit is a multi-initial object in the category of cocones. This must be a set of cocones, right?

Does a large discrete category have a multi-initial object?

Yes, multi-colimits must be indexed by a (cardinally) small collection, hence non-small discrete categories never have multi-initial objects.

Remark: multi-colimits of shape $\mathcal{S}$ can be regarded as a relative left adjoint of the diagonal functor $\mathcal{C} \to \mathcal{C}^\mathcal{S}$ with respect to the coproduct-cocompletion $\mathcal{C}\hookrightarrow \mathrm{Fam}(\mathcal{C})$.

[ykawase5048]

Then, the current description of multi-initial objects lacks the smallness condition on the collection of connected components. Sorry, this is my bad.

[ScriptRaccoon]

I see. Can you please make a small PR with the fix? I think the implications are not affected?

[ykawase5048]

Please find #118 well. I believe this change doesn't affect any implications.