# ind-completion vs cocompletion of a category, i.e Ind(C) vs Psh(C)

Let $C$ be any category. $Ind(C)$ is the category universal for filtered-colimit-preserving maps from $C$ to any inductive category (i.e categories having all filtered colimits), while the presheaf category $Psh(C):=Fun(C^{op}, Sets)$ is universal for colimit preserving maps from $C$ to a co-complete categories (i.e has all colimits). What are some examples where $Ind(C)$ and $Psh(C)$ are different?

I'm reading Lurie's Higher Topos theory, and where he talks about taking $C$ be the category of finitely generated abelian groups, and claims $Ind(C)$ equivalent to the category $Ab$ of all abelian groups. I think the reason is as follows: Given an abelian group $A$, we can consider the diagram of all finitely generated subgroups of $A$, with the maps being the inclusions map $A_i \to A_j$ if $A_i \subseteq A_j$. Then this is an $\omega$-filtered colimit, since any finite number of finitely generated subgroups has their union as an upper bound, and the union is also finitely generated. And $A\simeq colim_D \ A_i$. What is $Psh(C)$ in this case, and is it different from $Ab$?

• The universal property you describe for $\text{Psh}(C)$ is correct if $C$ is essentially small; otherwise you need to modify the construction of $\text{Psh}(C)$ to throw out some presheaves that are too big. Commented Oct 9, 2015 at 14:46
• Your argument shows that every abelian group is a filtered colimit of f.g. abelian groups, but this is not yet enough to show that the ind-completion of the latter is the former. Commented Oct 9, 2015 at 15:08
• @QiaochuYuan what else do i need to show? i guess i would have to something about hom sets, i.e every map between abelian groups is a colimit of maps between finitely generated submodules. But based on your comment I suspect I'm overlooking something else. Commented Oct 9, 2015 at 20:44

## 2 Answers

These almost never agree, in the sense that there's a natural functor $\text{Ind}(C) \to \text{Psh}(C)$ and it is almost never an equivalence. If $C$ is essentally small and has finite colimits, then $\text{Ind}(C)$ can be identified with the full subcategory of $\text{Psh}(C)$ on presheaves $F : C^{op} \to \text{Set}$ which send finite colimits to finite limits. An arbitrary presheaf just won't have this property. For example, if $C$ is the category of finitely generated abelian groups, the presheaf $\text{Hom}(-, \mathbb{Z}) \sqcup \text{Hom}(-, \mathbb{Z})$ does not have this property.

This should be a comment about possible notation confusion, but I don't have enough reputation for it.

If $$C$$ is a pre-triangulated small $$k$$-linear dg category (admitting finite limit and colimit), and Vect is the dg (derived) category of chain complexes over a field $$k$$, people sometimes write $$Mod(C) = Fun_{ex}(C^{op}, Vect)$$ as the dg category of exact functors. If we define Ind(C) as the ind-completion of the Yoneda image of $$C$$ in $$Mod(C)$$, then $$Ind(C)=Mod(C)$$, since $$Ind(C)$$ is closed under finite colimit and filtered colimit, hence is closed under all (small) colimit.

The possible confusion is that sometimes people write $$Psh(C)$$ as $$Mod(C)$$, and one need to watch for whether the functor category is valued in Set or Vect, and whether it is about exact functor or all functors.

(Note: the objects are not just the naive exact functors. For example, one can view dg cat as $$A_\infty$$ cat, then $$A_\infty$$ functors between two $$A_\infty$$ categories forms a $$A_\infty$$ category, and if the target category is dg, then the functor category is dg. (see Seidel's book on Fukaya category and Lefschetz fibration.) But I don't know how to unpackage / simplify this to the dg setting.)