homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
The notion of a regular semicategory generalizes the notion of a regular algebra from ring theory to (semi-)category theory. Just like regular algebras are (in general) non-unital algebras that nevertheless “behave” in many respects like unital ones, regular semicategories are semicategories on the verge of being categories.
Note that regular semicategories that are categories are not (necessarily) regular categories in the usual sense. In this case there is a clash of terminology between category theory and algebra.
For convenience let us first recall a couple of concepts
Let , be semicategories. A morphism of semicategories assigns to all objects an object and to every morphism in a morphism in such that .
A natural transformation consists of a family in indexed by the objects of such that for all in the following diagram commutes:
A presheaf on a semicategory is a morphism of semicategories . The category has objects presheaves on and morphisms the natural transformations and is the called the category of presheaves of the semicategory .
is indeed a category! Denoting the category resulting from by adding the missing identity morphisms, it is easy to check that and that the latter coincides with the usual presheaf category hence is even a Grothendieck topos.
Given there is also a Yoneda morphism defined on objects as usual by . Since semicategories embed into categories only iff they are categories themselves it follows that is fully-faithful iff is a category!
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The most striking result is that although for a general regular semicategory regular presheaves will not be Yoneda presheaves and vice versa nevertheless the subcategories and are identical in an identity-and-unity of opposites in the sense of Lawvere i.e. both are equivalent and occur in an essential localization of .
Let be a regular semicategory. The functor defined on objects by is fully-faithful and part of an adjoint string
with identifying the regular presheaves with the Yoneda presheaves and identifying them with the presheaves that are colimits of representables.
For the proof see Moens et al. (2002, p.179).
Regular semicategories were introduced in
Their quantaloid-enriched theory is studied in
For the origins in algebra of the concept see
Last revised on May 30, 2018 at 09:04:53. See the history of this page for a list of all contributions to it.