In general, let and be two families (indexed sets) of objects of some category . We say that is a refinement of if there is a function of indices and a morphism for each .
A common special case is the concept of refinement of open covers, example below.
We state a list of examples, beginning with general cases and then consecutively making them more specific.
Very often we do this in the slice category for some object . If you spell this out, then you have families and of morphisms to ; is a refinement of if there are a function and a commutative diagram
for each .
More specifically, apply this to the poset of subobjects of . Then you have families and of subobjects of ; is a refinement of if there are a function and a commutative diagram (1) for each .
Yet more specifically, apply this to the lattice of subsets of some set . Then you have families and of subsets of ; is a refinement of if there is a function such that each is contained in .
Yet more specifically, let the families of subsets be indexed by themselves. Then have collections and of subsets of ; is a refinement of if for each there is an such that is contained in .
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 there is an and a morphism .
(refinement of open covers)
Special cases of example include refinement of filters and refinement of open covers of topological spaces.
Let be a topological space, and let be a set of open subsets which covers in that .
Then a refinement of this open cover is a set of open subsets which is still an open cover in itself and such that for each there exists an with .
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 and of some object ; is a refinement of if there are a map and a commutative diagram (1) for each .
Refinement of open covers is a concept appearing in the definition of
Last revised on June 21, 2024 at 17:43:23. See the history of this page for a list of all contributions to it.