homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
A subcategory of a strict category is called replete if it respects isomorphism of morphisms in the arrow category of . It is a subcategory for which the property of (strictly) belonging to it respects the principle of equivalence of categories.
A subcategory of is replete if for any object in and any isomorphism in , both and are also in . Equivalent ways to state this include:
If and in the arrow category , then .
The inclusion is an isofibration.
Since repleteness is a “closure condition,” the intersection of any collection of replete subcategories is again replete. Therefore, any subcategory is contained in a smallest replete subcategory, called its repletion. We can construct the repletion of explicitly as follows:
Repleteness and repletions are most often applied to full subcategories, in which case the repletion is simply the full subcategory of determined by those objects which are isomorphic to some object of . In particular, in this case, the repletion is equivalent to . More generally, we can say:
The inclusion is an equivalence if and only if the inclusion is pseudomonic.
The inclusion of a replete subcategory is always pseudomonic, so if is an equivalence, and in particular full and faithful, then must also be pseudomonic.
Conversely, if is pseudomonic, then every morphism of can be written as for some morphism in and isomorphisms and in . If the domain and codomain are in , then since is pseudomonic, both and must also be in , hence such a morphism is itself necessarily in . Thus the inclusion is full. It is clearly faithful and essentially surjective, so it is an equivalence.
The replete image of a functor is the repletion of its image. The replete full image of a functor is the repletion of its full image, i.e. the full subcategory of its target determined by those objects isomorphic to some object in its image. See also essential image.
More generally, for , a subcategory of an -category is replete if all equivalences in whose either source or target is in are themselves in . In particular, all -cells equivalent in to some -cell in are themselves in (because they are sources of some equivalence in ). Then replete subcategories of -categories are those which are replete in this sense.
Here we are using the weakest notion of equivalence; one could also talk about -replete -subcategories if every -equivalences in belongs to if either its source or target does. Then using the usual definition of ‘category’ as a model for -categories, replete subcategories are already those which are -replete (indeed, -equivalence is isomorphism in this model); but using a model of -categories in which hom-sets are setoids (as is common, for example, in type-theoretic foundations), we must insist on -repleteness.
Note that a subcategory of an -category is -replete if and only if:
Last revised on September 14, 2024 at 10:35:39. See the history of this page for a list of all contributions to it.