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Created arity class. Added links from a few places, but there are probably others I didn’t think of.
Excellent term! Is it yours?
Yes. I’m glad you like it!
Is the full subcategory spanned by a class of arities always also a category of arities for the identity monad, in the sense of a monad with arities? This is certainly true for $\{ 1 \}$ (for trivial reasons), $\aleph_0$ (because filtered colimits preserve finite products), and $\infty$ (because the diagram in question has a terminal vertex). But I’m less certain about the other regular cardinals, especially the finite ones…
Good question! Let’s see, $Set$ is locally $\kappa$-presentable for any infinite regular cardinal $\kappa$, so the $\kappa$-small sets are $\kappa$-presentable and the associated canonical colimits are $\kappa$-filtered. Seems like that’s exactly what we need to conclude that the identity monad has the appropriate arities.
For the unique finite regular cardinal $2$, the $2$-small sets are 0 and 1, so we only need to worry about whether mapping out of 0 preserves the canonical $2$-colimits. Being 2-filtered means being inhabited, but mapping out of 0 doesn’t preserve inhabited colimits (e.g. coproducts). It does, however, preserve connected colimits, and any canonical colimit w.r.t. 2 is connected (the diagram has an initial object). So I think we’re good there too.
That suggests that maybe there’s some sense in which “$2$-filtered” ought to mean “connected”? I can’t see how that could be true other than by convention, though. Maybe I’m just confused.
I’m quite confused as well: as far as I can tell, the only reason we want $\nu_{\mathcal{A}} T$ to preserve canonical colimits is so that Beck–Chevalley condition is satisfied for the obvious diagram of functors, but in the case $T = id$ this is automatic… (Assuming I haven’t made any mistakes, the density of $\Theta \hookrightarrow \mathcal{C}^\mathbb{T}$ can be proven with weaker assumptions.)
Regarding 2-filteredness: if we (re)define “small $\kappa$-filtered category $\mathcal{J}$” to mean “$colim_\mathcal{J}$ preserves $\kappa$-limits in $\mathbf{Set}$”, then a 2-filtered category is precisely a connected inhabited category. Since $\kappa$-filtered categories in the traditional sense are connected for $\kappa \ge \aleph_0$, this doesn’t seem to be too much to ask for…
the only reason we want $\nu_{\mathcal{A}} T$ to preserve canonical colimits is so that Beck–Chevalley condition is satisfied for the obvious diagram of functors, but in the case $T = id$ this is automatic…
Well, sure, but that’s separate to the question of whether the arity class gives arities to the identity monad for the formal definition of “has arities”, right? It’s just saying that in the special case of Id, the formal definition of “has arities” is stronger than necessary for the conclusion of the main general theorem about it to hold.
if we (re)define “small $\kappa$-filtered category $\mathcal{J}$” to mean “$colim_\mathcal{J}$ preserves $\kappa$-limits in $\mathbf{Set}$”, then a 2-filtered category is precisely a connected inhabited category.
Ah, yes!! And that’s the right definition, too, because it generalizes even further.
(Small point: a connected category is automatically inhabited.)
a connected category is automatically inhabited
Smile. I expect Zhen was just being on the safe side, since there remain some people who think empty things are connected. :-)
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