nLab
refinement

Contents

Idea

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

A common special case is the concept of refinement of open covers, example 4 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/XC/X for some object XX. If you spell this out, then you have families (U iX) i(U_i \to X)_i and (V jX) j(V_j \to X)_j of morphisms to XX; VV is a refinement of UU if there are a function f:JIf: J \to I and a commutative diagram

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

for each jj.

Example

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

Example

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

Yet more specifically, let the families of subsets be indexed by themselves. Then have collections U𝒫XU \subseteq \mathcal{P}X and V𝒫XV \subseteq \mathcal{P}X of subsets of XX; VV is a refinement of UU if for each jj there is an ii such that V jV_j is contained in U iU_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 jj there is an ii and a morphism V jU iV_j \to U_i.

Example

(refinement of open covers)

Special cases of example 3 include refinement of filters and refinement of open covers of topological spaces.

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

Then a refinement of this open cover is a set of open subsets {V jX} jJ\{V_j \subset X\}_{j \in J} which is still an open cover in itself and such that for each jJj \in J there exists an iIi \in I with V jU iV_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 iX) i(U_i \to X)_i and (V jX) j(V_j \to X)_j of some object XX; VV is a refinement of UU if there are a map f:JIf: J \to I and a commutative diagram (1) for each jj.

Examples

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

Revised on April 27, 2017 11:27:45 by Urs Schreiber (131.220.184.222)