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Stream: theory: category theory

Topic: categorification of way-below from domain theory

Kenji Maillard (Jun 03 2020 at 17:16):

I'm used to think of many elements of domain theory as a "simplification" of what happens in general CT where we only consider posets (so (0,1)-categories). For instance algebraic dcpos are the domain theoretic analog of $(\omega$-)accessible categories. I realized recently that I didn't know any categorical analog to the way-below relation on elements of a dcpo.
Here is a tentative translation: let $\mathcal{C}$ be a category with filtered colimits, a way-below structure on a morphism $f \in \mathcal{C}(x,y)$ is the data for any filtered diagram $\mathcal{F} : \mathcal{I} \to \mathcal{C}$ of a function $h_\mathcal{F} : \mathcal{C}(y, \mathrm{colim}_i\,\mathcal{F}\,i) \to \mathrm{colim}_i\, \mathcal{C}(x, \mathcal{F}\,i)$ filling the span

$\mathcal{C}(y, \mathrm{colim}_i\,\mathcal{F}\,i) \xleftarrow{} \mathrm{colim}_i\, \mathcal{C}(y, \mathcal{F}\,i) \xrightarrow{-\circ f} \mathrm{colim}_i\, \mathcal{C}(x, \mathcal{F}\,i)$

(I guess $h_\mathcal{F}$ should be natural in $\mathcal{F}$ -- and also in the indexing filtered category $\mathcal{I}$)

There is a clear action by precomposition by a morphism $g \in \mathcal{C}(x',x)$ so this definition induces a "presheaf" on $\mathcal{C}_{/y}$ (up to possible size issues).
Is that object known ? Studied ?

Dan Doel (Jun 03 2020 at 17:23):

There's actually mention of this on nlab. E.G. on the continuous category page.

Dan Doel (Jun 03 2020 at 17:25):

I'm not sure if what you described is exactly what they define there or not.

Kenji Maillard (Jun 03 2020 at 18:01):

The nlab page looks quite relevant, in particular the But unlike in the posetal case, it is not clear how to define wavy arrows unless CC is continuous (whereas ≪\ll can be defined in any poset with directed joins). part, thanks a lot for the link!

Dan Doel (Jun 03 2020 at 18:04):

No problem.

Morgan Rogers (he/him) (Jun 04 2020 at 09:49):

See also chapter C4 of the Elephant, which includes topos-theoretic motivation for this generalisation.