nLab refinement

Idea

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

A common special case is the concept of refinement of open covers, example below.

Examples

We state a list of examples, beginning with general cases and then consecutively making them more specific.

Example

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 $j$.

Example

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 $j$.

Example

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 $V_j \to U_i$.

Example

(refinement of open covers)
Special cases of example include refinement of filters and refinement of open covers of topological spaces.

Let $(X,\tau)$ be a topological space, and let $\{U_i \subset X\}_{i \in I}$ be a set of open subsets which covers $X$ in that $\underset{i \in I}{\cup} U_i \;= \;X$.

Then a refinement of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an open cover in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

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 $j$.

Examples

Refinement of open covers is a concept appearing in the definition of

• compact topological space, paracompact topological space, etc.

Related concepts

• Čech homology