Question

This question probably belongs to the very basics of Category Theory, but I have not found an appropriate answer in the latest hours.

Suppose that one has a category $\mathcal{C}$ in which direct limits exist and $\mathcal{I}$ and $\mathcal{K}$ are directed sets. By a Double Direct System in $\mathcal{C}$ (w.r.t $\mathcal{I}$ and $\mathcal{K}$), {$A_{i}^{k},f_{ji}^{lk}$} ($i\in \mathcal{I}, k\in \mathcal{K}$), I will mean a collection of objects $A_{i}^{k}\in \mathcal{C} $ and, for each $i\leq j$ and $r\leq s $, morphisms $f_{ji}^{sr}:A_{i}^{r}\rightarrow A_{j}^{s}$ such that:

1. $f_{ii}^{rr}=Id:A_{i}^{r}\rightarrow A_{i}^{r}$.

2. The composite $A_{i}^{r}\stackrel{f_{ji}^{sr}}{\rightarrow}A_{j}^{s}\stackrel{f_{kj}^{ts}}{\rightarrow}A_{k}^{t}$ equals $f_{ki}^{tr}$.

Note that, fixing $r$, one gets a direct system {$A_{i}^{r}, f_{ji}^{rr}$}, and there is, for each $r\leq s$, an induced map $$f^{sr}:\varinjlim_{i}A_{i}^{r}\rightarrow \varinjlim_{i}A_{i}^{s}$$ which makes {$\varinjlim_{i}A_{i}^{r}, f^{sr}$} a direct system.

Repeating this procedure in the subindices one gets a direct system {$\varinjlim_{r}A_{i}^{r}, f_{ji}$}. My question is: under which conditions (over $\mathcal{C}$, or over the involved morphisms) does one has a (unique?) isomorphism $$\varinjlim_{i}\ \varinjlim_{r}A_{i}^{r}\cong\varinjlim_{r} \ \varinjlim_{i}A_{i}^{r}$$ ?

Remarks:

1. Instead of the given definition one can assume the (apparently?) weaker condition that, fixing subindices $i$, the system {$A_{i}^{r},f_{ii}^{sr}$}$_{r,s\in\mathcal{K}}$ is direct, and analogously for superindices. What do one gets in this case?

2. There is a theorem about "Interchange of Limits" in Mac Lane's "Categories for the Working Mathematician", but I do lack of a proper background to quickly see whether this answers my question. Should I do the effort of understanding that result for this?

3. (Just for information) The question did arise while I was solving Ex. II.1.10 from Hartshorne's "Algebraic Geometry", as my solution carried me to check the validity of the equation $$(\varinjlim_{i} \mathcal{F}_{i})_P\cong \varinjlim_i (\mathcal{F}_i)_P$$

where $\mathcal{F}_{i}$ is a direct system of sheaves of abelian groups on a topological space $X$. In this case I could achieve an isomorphism using the universal property for direct limits of abelian groups and using constant sheaves on $X$, but this seems to be quite restrictive.

Answer 1

Your indices are hard to read, and I think you might have mixed some of them up.

If you mean what I think you mean, then Mac Lane's result is exactly what you need. The two colimits are canonically isomorphic to each other and to the colimit over the product category $\mathcal{I} \times \mathcal{K}$.

If you assume only that the system is functorial in $i \in \mathcal{I}$ and $k \in \mathcal{K}$ separately, then what you have is not necessarily a functor $\mathcal{I} \times \mathcal{K} \to \mathcal{C}$ (it's what John Power and Edmund Robinson call a binoidal functor), and I don't know if there is any sensible notion of limit for such a thing.

Answer 2

I think you will find an answer in par. 2.4 of this

http://people.math.jussieu.fr/~schapira/lectnotes/AlTo.pdf

In particular I think one always have $$ \varprojlim_I\;\varprojlim_J A_{ij} \cong \varprojlim_J\; \varprojlim_I A_{ij} $$ because $\varprojlim$ is right adjoint in an adjoint couple, and right adjoints preserve projective limits...

Answer 3

Ler $I$ and $J$ graphs (like small categories, but whitout compositions and identity) and let $\mathscr{C}$ a category by colimits respect to $I$ and $J$. Then if $\mathscr{C}$ as $I\times J$ colimits and for $F: I\times J\to \mathscr{C} $ the two partial limits commute and their compositions is the limit of $F$ :

$colim_{i\in I} (lim_{j\in J} F(i, j)) \cong colim_{(i, j)\in I\times J} F(i, j) \cong colim_{j\in J} (colim_{i\in I} F(i, j)) $.

The some is true for limt (for duality)

PROOF. Let $F_I(j):= {\underrightarrow{lim}}_{i\in I} F(i, j)$ follow $F_I: J\to \mathscr{C}: j \mapsto F_I(j)$, the first equivalence mean that the colimits of $F_I$ is the colimit of $F$, this folow because a cocone $f_j: F_I(j)\to X,\ j\in J$ as a unique extention (considering the colimits cocones $\varepsilon_{i,j}: F(i, j)\to F_I(j),\ i\in I$) to a cocone $f_{i,j}=f_j\circ \varepsilon_{i, j}: F(i,j)\to X,\ (i,j)\in I\times J$. The second equivalence is similar.

Then your assert is true for general $I$ and $J$ but if $I=J$ is directed (or in general a filtrant category) then $lim_{I\times I} F\cong lim_{i\in I} F(i,i)$.

The more interesting propriety is the follow: if $I $ is a filtrant small category (in particolar a directed order) and $J$ a finite graph then for $F: I\times J\to \mathscr{C} $ and $\mathscr{C}$ is locally finitely-presentable then $I$-colimits and $J$-colimits commute:

$colim_{i\in I} lim_{j\in J} F(i,j) \cong lim_{J} colim_{I} F(i, j) $.