# Ind-objects in a coherent category

For a poset $P$, an ideal of $P$ is a downward closed upward filtered set of elements of $P$. The collection of ideals of $P$ ordered by inclusion is the free cocompletion of $P$ w.r.t. filtered colimits. If indeed the poset $P$ is a distributive lattice then one can simplify this definition a bit and define an ideal equivalently as a downward closed set which is closed under finite joins.

The appropriate generalization from posets to categories is that of ind-objects. An ind-object of $C$ is a diagram $D \to C$ where $D$ is a small filtered category. The free cocompletion of $C$ w.r.t. filtered colimits, denoted $Ind\text{-}C$ has ind-objects of $C$ as objects and for $F:D \to C$ and $G:E \to C$ two such ind-objects, the hom-set $Ind\text{-}C(F, G)$ is defined as $\underset{d\in D}{lim}\; \underset{e\in E}{colim}\; C(F d, G e)$ with limits and colimits computed in $Set$.

Now I need to know if there is any kind of simplification in case the category $C$ happens to be coherent. If we equivalently define an ind-object as a filtered colimit of representable presheaves there is a simplification: $Ind\text{-}C$ will be the full subcategory of PSh(C) consisting of those functors $F:C^{op} \to Set$ such that $F$ is left exact and satisfies a certain smallness requirement. So is there too a simplification for the definition of $Ind\text{-}C$ given above?

(My naive guess was to take as objects those diagrams $F:D \to C$ where $D$ has finite colimits and $F$ preserves them, but this obviously does not work.)

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Maybe Chapter 4 of Deligne's paper "le groupe fondamental de la droite projective moins trois points" might help. See math.ias.edu/files/deligne/GaloisGroups.pdf – Harry May 4 '12 at 9:07
$Ind(C)$ is equivalent to the left exact presheaves (with a genrating set on its comma categy) if $C$ has finite limits. – Buschi Sergio May 4 '12 at 12:04
$Ind(C)$ is equivalent to the full subcategory $A\subset C^>$ of presheaves F such that the comma category $C\downarrow F$ is filtrant and with a final set of objects. (see S. Mardešić, J. Segal, Shape theory, North Holland 1982.) – Buschi Sergio May 4 '12 at 12:15
Thanks Buschi, that is helpful but additionally I want to know what kind of diagrams they correspond to. Is there any relation with the diagram being flat as a functor $J^{op} \to C^{op}$ or something like that? – Ali Lahijani May 4 '12 at 17:59
From Sheaves in geometry and logic - MAc LAne MOerdijk, these are exatly the flat presheaves (see p. 386) (if $C$ is small) – Buschi Sergio May 5 '12 at 13:23