refinement

In general, let $U = (U_i)_{i: I}$ and $V = (V_j)_{j: J}$ be two families of objects of some category $C$. We say that $V$ is a **refinement** of $U$ if there are a function $f: J \to I$ of indices and a morphism $V_j \to U_{f(j)}$ for each $j$.

Very often we do this in the slice category $C/X$ for some object $X$. If you spell this out, then you have families $(U_i \to X)_i$ and $(V_j \to X)_j$ of morphisms to $X$; $V$ is a refinement of $U$ if there are a function $f: J \to I$ and a commutative diagram

(1)$\array{
V_j &&\to&& U_{f(j)}
\\
& \searrow && \swarrow
\\
&& X
}$

for each $j$.

More specifically, apply this to the poset of subobjects of $X$. Then you have families $(U_i \hookrightarrow X)_i$ and $(V_j \hookrightarrow X)_j$ of subobjects of $X$; $V$ is a refinement of $U$ if there are a function $f: J \to I$ and a commutative diagram (1) for each $j$.

Yet more specifically, apply this to the lattice of subsets of some set $X$. Then you have families $(U_i \subseteq X)_i$ and $(V_j \subseteq X)_j$ of subsets of $X$; $V$ is a refinement of $U$ if there is a function $f: J \to I$ such that each $V_j$ is contained in $U_{f(j)}$.

Yet more specifically, let the families of subsets be indexed by themselves. Then have collections $U \subseteq \mathcal{P}X$ and $V \subseteq \mathcal{P}X$ of subsets of $X$; $V$ is a refinement of $U$ if for each $j$ there is an $i$ such that $V_j$ is contained in $U_i$.

Actually, this definition is slightly weaker than the previous one in the absence of the axiom of choice. Perhaps in that case the general definition should say that for each $j$ there is an $i$ and a morphism $V_j \to U_i$.

Special cases of this last example include refinement of filters and refinement of open covers.

On the other hand, you might want to generalise the case of open covers to covers or covering sieves on a site. In that case, the general definition still applies; you have covering families $(U_i \to X)_i$ and $(V_j \to X)_j$ of some object $X$; $V$ is a refinement of $U$ if there are a map $f: J \to I$ and a commutative diagram (1) for each $j$.

Revised on July 19, 2009 20:24:22
by Toby Bartels
(71.104.230.172)