9.6 Strict categories
Definition 9.6.1.
A strict categoryis a precategory whose type of objects is a set.
In accordance with the mathematical red herring principle, a strict category is not necessarily a category.In fact, a category is a strict category precisely when it is gaunt (\\autorefct:gaunt).Most of the time, category theory
is about categories, not strict ones, but sometimes one wants to consider strict categories.The main advantage of this is that strict categories have a stricter notion of “sameness” than equivalence, namely isomorphism
(or equivalently, by \\autorefct:cat-eq-iso, equality).
Here is one origin of strict categories.
Example 9.6.2.
Let be a precategory and an object.Then there is a precategory as follows:
- •
Its objects consist of an object and a monomorphism
.(As usual, is a monomorphism (or is monic) if .)
- •
Its morphisms from to are arbitrary morphisms from to in (not necessarily respecting and ).
An equality of objects in consists of an equality and an equality , which by \\autorefct:idtoiso-trans is equivalently an equality .Since hom-sets are sets, the type of such equalities is a mere proposition.But since and are monomorphisms, the type of morphisms such that is also a mere proposition.Thus, if is a category, then is a mere proposition, and hence is a strict category.
This example can be dualized, and generalized in various ways.Here is an interesting application of strict categories.
Example 9.6.3.
Let be a finite Galois extensionof fields, and its Galois group.Then there is a strict category whose objects are intermediate fields , and whose morphisms are field homomorphisms which fix pointwise (but need not commute with the inclusions into ).There is another strict category whose objects are subgroups , and whose morphisms are morphisms of -sets .The fundamental theorem of Galois theory
says that these two precategories are isomorphic (not merely equivalent
).