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Let $F:C \to D$ be a full and faithful functor between small categories. Then we get a triple of adjoint functors $F_! \dashv F^* \dashv F_*$, with $$F_!:Set^{C^{op}} \to Set^{D^{op}}.$$

Notice that $F_!=Lan_{y_C} \left(y_D \circ F\right),$ where in both cases $y$ denotes the Yoneda embedding, so that $F_!$ is left-exact if and only if $y_D \circ F$ is filtering. (Remark: I do NOT want to assume that C has finite limits, since it doesn't in my example,so filtering $\ne$ left-exact).

I'm looking for a stronger statement however. Suppose that $y_D \circ F$ is NOT filtering, so that $F_!$ is NOT left-exact. Nonetheless, $F_!$ may preserve certain finite limits (perhaps those in the image of a certain left-exact functor etc.). My question is, can one characterize (or give a sufficient condition for) those limits in $Set^{C^{op}}$ which ARE preserved by $F_!$?

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In the paper A classification of accessible categories by Adámek, Borceux, Lack, and Rosický, they prove that if $\mathbb{D}$ is a collection of small categories satisfying a technical condition called soundness, then the following are equivalent for a functor $F\colon C\to Set$:

  1. $\mathrm{Lan}_{y_C} F : \mathrm{Set}^{C^{\mathrm{op}}} \to \mathrm{Set}$ preserves $\mathbb{D}$-limits.
  2. $\mathrm{el}(F)^{\mathrm{op}}$ is $\mathbb{D}$-filtered, i.e. its category of cocones under any $\mathbb{D}$-diagram is connected.

Applying this objectwise to a functor $F\colon C\to D$, you can recover a condition, which I would call "representably $\mathbb{D}$-flat", which is equivalent to $F_!$ preserving $\mathbb{D}$-limits.

Thus, one sufficient condition for a particular limit in $\mathrm{Set}^{C^{\mathrm{op}}}$ to be preserved by $F_!$ is that it is a $\mathbb{D}$-limit for some sound $\mathbb{D}$ for which $F$ is representably $\mathbb{D}$-flat.

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Thanks Mike. This is interesting, and fits into the scope of the question the way I asked it. I took a look at the paper however, and it appears it does not help me in my situation. Unless I am misunderstanding things, a $mathbf{D}$-limit is a limit of a certain "shape," i.e. the limit of ANY functor with domain in $\mathbf{D}$ is called a $\mathbf{D}$-limit. In my case, I actually have a distinguished class of equalizers and I want to know if they are preserved, so their shape is really governed by the doctrine $FIN$- but it's the actual functor that realizes them that I want control. –  David Carchedi May 8 '12 at 22:51
    
I.e., I know that not all finite limits are preserved, but, I expect that when the diagram presenting a finite limit has nice enough conditions, that it may still be preserved, and I want to pin down these conditions, or at least find some sufficient ones that are satisfied in my situation. –  David Carchedi May 8 '12 at 22:53
    
Yes, I wasn't sure that this would help you, but I thought it was worth mentioning. (There are sound doctrines smaller than FIN which contain equalizers, though, like the doctrine of finite connected limits. I would guess that the doctrine containing only equalizers is probably not sound.) –  Mike Shulman May 9 '12 at 6:01
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