A map between direct limits

Question

Let $C$ be a category which has all small colimits.

I have the following situation: $\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$ are two directed systems in $C$, with transition maps $\alpha_{i_1,i_2}$ and $\beta_{j_1,j_2}$ respectively.

Assume that $\sigma: I \to P(J)$ is a map of sets (where $P(J)$ is the collection of subsets of $J$), such that $\sigma(i) \ne \emptyset$ for all $i \in I$. Moreover, if $j \in \sigma(i)$, and $k\ge j$, then $k \in \sigma(i)$.

For any $i \in I$, and any $j \in \sigma(i)$, I have a morphism $\gamma_{i,j}: A_i \to B_j$. These are compatible with the transition maps: if $j,k \in \sigma(i)$, and $j\le k$, then $\beta_{j,k} \circ \gamma_{i,j} = \gamma_{i,k}$.

Also, the image of $\sigma$ covers $J$, that is $\cup_{i \in I} \sigma(i) = J$.

My question is: can I, in such a situation, obtain a natural map $\varinjlim_{i \in I} A_i \to \varinjlim_{j \in J} B_j$?

If I had $I = J$ then this would be simply the functoriality of colimits, as the maps $\gamma$ will give rise to a morphism of directed systems. But can I construct such a map in this more general situation?

Answer 1

Let $A = \varinjlim_{i \in I} B_i$, $B = \varinjlim_{j \in J} B_j$, and for each $j \in J$, let $\beta_{j\infty}: B_j \to B$ be the colimit inclusion (i.e. the leg at $j$ of the colimiting cocone). Define, for each $i \in I$, $\gamma_i : A_i \to B$ to be $\beta_{j\infty}\gamma_{ij}$ for some $j \in \sigma(i)$. It's easy to check, using directedness of $J$ and the upward closedness of $\sigma(i)$, that this definition is independent of the choice of $j$. The claim is that $(\gamma_i)_{i \in I}$ forms a cocone from $(A_i)_{i \in I}$ to $B$, and you want the induced morphism $\gamma: A \to B$.

In order for $(\gamma_i)_{i\in I}$ to be a cocone, you need naturality in $i$, and to get this you probably want some naturality of $\gamma_{ij}$ in $i$ (you've only mentioned a sort of naturality in $j$). Namely, if $i \leq i'$ and $j \in \sigma(i) \cap \sigma(i')$ (there always is such a $j$ in your setup), hopefully you have

($\ast$) $\gamma_{i'j}\alpha_{ii'} = \gamma_{ij}$.

If ($\ast$) holds, then it's easy to check that $(\gamma_i)_{i \in I}$ forms a cocone and you get an induced morphism $\gamma: A \to B$ as desired. The fact that $J = \cup_{i \in I} \sigma(i)$ turns out not to be necessary.

I suppose it would suffice to check, for all $i \in I$, that ($\ast$) holds only for some $j \in \sigma(i)$ (equivalently, for a cofinal set of $j \in \sigma(i)$). For that matter, ($\ast$) is not strictly necessary -- you can simply ask that $\gamma_{i'} \alpha_{ii'} = \gamma_i$, straight up. But ($\ast$) seems a fairly natural thing to check for.