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 $F_!$?


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.

  • $\begingroup$ 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. $\endgroup$ – David Carchedi May 8 '12 at 22:51
  • $\begingroup$ 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. $\endgroup$ – David Carchedi May 8 '12 at 22:53
  • $\begingroup$ 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.) $\endgroup$ – Mike Shulman May 9 '12 at 6:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.