refinement (changes) in nLab

Context

Category theory

• Idea
• Examples
• Examples
• Related concepts

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 i→X) i(U_i \to X)_i and (V j→X) j(V_j \to X)_j of morphisms to XX; VV is a refinement of UU if there are a function f:J→If: 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 i↪X) i(U_i \hookrightarrow X)_i and (V j↪X) j(V_j \hookrightarrow X)_j of subobjects of XX; VV is a refinement of UU if there are a function f:J→If: 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 i⊆X) i(U_i \subseteq X)_i and (V j⊆X) j(V_j \subseteq X)_j of subsets of XX; VV is a refinement of UU if there is a function f:J→If: 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 U⊆𝒫XU \subseteq \mathcal{P}X and V⊆𝒫XV \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.

Example

(refinement of open covers ) \linebreak Special cases of example3 include refinement of filters and refinement of open covers of topological spaces.

Special Let cases of example3(X,τ)(X,\tau) include be refinement a of filters topological space , and refinement let ofopen covers{U i⊂X} i∈I\{U_i \subset X\}_{i \in I} be a set of topological open spaces subsets which covers XX in that ∪i∈IU i=X\underset{i \in I}{\cup} U_i \;= \;X.

Let Then a(X,τ)(X,\tau)refinement be of this open cover is a set of open subsetstopological space{V j⊂X} j∈J\{V_j \subset X\}_{j \in J} , and which let is still an{U i⊂X} i∈I\{U_i \subset X\}_{i \in I}open cover be in a itself set and of such that for eachopen subsetsj∈Jj \in J which there exists ancoversi∈Ii \in I withXV j⊂U i X V_j \subset U_i in that ∪i∈IU i=X\underset{i \in I}{\cup} U_i \;= \;X.

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