On the definition of "exact filtered colimits"

[ykawase5048]

The implication finitely accessible" ==> "filtered-colimit-stable monos is discussed in #297 (comment).
I think this would be a corollary of the following more general fact: in any κ-accessible category $A$, κ-filtered colimits commute with existing κ-limits.
This follows from the properties below satisfied by the canonical embedding $A\to Set^{A_{κ}^{op}}$ to the presheaf category over the full subcategory of κ-presentables:

• It is closed under κ-filtered colimits.
• It preserves any existing limits in $A$.
• It reflects limits.

This suggests us to remove the assumption on the existence of finite limits (perhaps also filtered colimits?) from the definition of exact filtered colimits.

I also think it would be best to cite a source that explicitly states this basic fact, but I haven't been able to find one for now.

[ScriptRaccoon]

The problems of removing finite limits or filtered colimits from "exact filtered colimits":

1. Which property do we remove? And why? (If one, why not the other?)
2. The equivalent formulations of the property do not hold anymore*.
3. We might end up having three different non-equivalent properties (all of which need a separate entry).
4. (lazy and weak point, but still want to mention it) We need to decide the property for lots of categories which are currently decided.

*In general, what does "limits of shape I commute with colimits of shape J" mean? Do we start with a bifunctor? What are the exact assumptions?

Similar remarks apply to other properties where other properties are assumed.

[ScriptRaccoon]

The discussion also applies to "filtered-colimit-stable monos" which currently requires filtered colimits to exist, but one might argue again to remove this assumption. Here I definitely agree, and only the "lazy point" remains. We then need to decide the property for 23 categories where it is currently decided.

And of course, the dual properties are affected as well.

[ykawase5048]

@ScriptRaccoon
The main reason I suggested removing the assumption of finite limits is actually much more concrete.

When I looked at the list of categories satisfying finitely accessible ∧ ¬exact filtered colimits in CatDat, I noticed that several categories appear there simply because they do not have finite limits. For example, the category of fields is on the list, even though the basic fact "pullbacks commute with filtered colimits in the category of fields" is still true. From a mathematical point of view, this feels rather weird to me, independently of any structural considerations about the database.

On the other hand, I only mentioned removing the assumption of filtered colimits for consistency. In fact, I am not particularly convinced that we should do that, since I don't have any motivating example for now.

Remark. If we were to weaken the assumptions in the way I suggested, I think the three equivalent definitions currently used on the exact filtered colimits page would only need minor adjustments: in (1), the domain category would have to be restricted, while in (2) and (3), one would simply add "whenever the (co)limits exist."