Given a presheaf $F: \mathcal{A} \to \text{Set}$, a classical result shows a connection between the limits preserved by $F$, the limits preserved by its Yoneda extension, and a property of the category of elements of $F$:
TFAE:
- $\text{Lan}_{y}F=:\text{Yan}(F)$ preserves finite limits;
- $\text{Elts}(F)^{\text{op}}$ is cofiltered.
If $\mathcal A$ has finite limits, this is in turn equivalent to
- $F$ preserves finite limits.
This classical result was significantly improved in [1], unveiling the connection between the shapes of limits $D\in\mathsf{Shapes}$ preserved by $F$ (and $\text{Yan}(F)$) and the fact that the category of elements of $F$ belongs to the "sound doctrine" (see [1] for the terminology) of categories $J$ such that colimits over $J$ commute with limits over each $D\in\mathsf{Shapes}$:
[1, Thm. 2.4] Let $\mathbb{D}$ be sound a doctrine, then TFAE for a functor $F: \mathcal{A} \to \text{Set}$ with a small domain:
- $\text{Lan}_{y}(F)$ preserves $\mathbb{D}$-limits of representables;
- $\text{Lan}_{y}(F)$ preserves $\mathbb{D}$-limits;
- $F$ is a $\mathbb{D}$-filtered colimit of representables;
- $\text{Elts}(F)^{\text{op}}$ is $\mathbb{D}$-filtered.
Moreover, if $\mathcal{A}$ is $\mathbb{D}$-complete, these condition are equivalent to:
- F is $\mathbb{D}$-continuous.
This result seems to suggest that there is a dictionary translating properties $\mathcal P$ of $F$ into properties $\mathcal P'$ of $\text{Yan}(F)$, and properties $\mathcal P''$ of $Elts(F)$; of course there is no hope to answer a question like "Given the generic property $\mathcal P$, determine $\mathcal P',\mathcal P''$".
Nonetheless, I hope that at least for some easy choices of $\mathcal P$, something can be said:
- Q1: What is a necessary and sufficient condition on $\text{Yan}(F)$ so that $F$ commutes with $\mathbb D$-colimits? What is a similar necessary and sufficient condition on $Elts(F)$?
- Q2: What is a necessary and sufficient condition on $\text{Yan}(F)$ so that $F$ is conservative? What is a similar necessary and sufficient condition on $Elts(F)$?
- Q3: What is a necessary and sufficient condition on $\text{Yan}(F)$ so that $F$ is faithful? What is a similar necessary and sufficient condition on $Elts(F)$?
As for Q1, one can attempt to blindly dualize the above statement obtaining:
Claim: Let $\mathbb{D}$ be sound a doctrine, then TFAE for a functor $F: \mathcal{A} \to \text{Set}$ with a small domain:
- $\text{Ran}_{y}(F)$ preserves $\mathbb{D}$-colimits of representables;
- $\text{Ran}_{y}(F)$ is $\mathbb{D}$-cocontinuous;
- $F$ is a $\mathbb{D}$-cofiltered limit of representables;
- $\text{Elts}(F)$ is $\mathbb{D}$-filtered.
Moreover, if $\mathcal{A}$ is $\mathbb{D}$-complete, these condition are equivalent to:
- F is $\mathbb{D}$-cocontinuous.
and going through the proof of [1, Thm. 2.4] everything seems to dualize.
I have no clue about which properties of $\text{Yan}(F)$ characterize the fact that $F$ is faithful or conservative. What is obvious is that if $\text{Yan}(F)$ is faithful or conservative, then so must be $F$ because $\text{Yan}(F)\circ y\cong F$ and $y$ is (full and) faithful.
General questions on how it's worth to pursue this track:
- Do these issues already appear in the literature? Can you point me any related reference?
- I have applications in mind for such a dictionary between properties of $F$ and properties of $\text{Yan}(F)$: do you think it's an interesting result, and worth to be published? Be honest!
Ref:
[1] Jiřı́ Adámek, Francis Borceux, Stephen Lack, Jiřı́ Rosický, A classification of accessible categories, In Journal of Pure and Applied Algebra, Volume 175, Issues 1–3, 2002, Pages 7-30, ISSN 0022-4049
