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 $A$ be a precategory and $x : A$ an object.Then there is a precategory $\\mathsf{mono}(A,x)$ as follows:

•
Its objects consist of an object $y : A$ and a monomorphism $m:\\hom_{A}(y,x)$ .(As usual, $m:\\hom_{A}(y,x)$ is a monomorphism (or is monic) if $(m\\circ f=m\\circ g)\\Rightarrow(f=g)$ .)
•
Its morphisms from $(y,m)$ to $(z,n)$ are arbitrary morphisms from $y$ to $z$ in $A$ (not necessarily respecting $m$ and $n$ ).

An equality $(y,m)=(z,n)$ of objects in $\\mathsf{mono}(A,x)$ consists of an equality $p:y=z$ and an equality ${p}_{*}\\mathopen{}\\left({m}\\right)\\mathclose{}=n$ , which by \\autorefct:idtoiso-trans is equivalently an equality $m=n\\circ\\mathsf{idtoiso}(p)$ .Since hom-sets are sets, the type of such equalities is a mere proposition.But since $m$ and $n$ are monomorphisms, the type of morphisms $f$ such that $m=n\\circ f$ is also a mere proposition.Thus, if $A$ is a category, then $(y,m)=(z,n)$ is a mere proposition, and hence $\\mathsf{mono}(A,x)$ 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 $E/F$ be a finite Galois extensionof fields, and $G$ its Galois group.Then there is a strict category whose objects are intermediate fields $F\\subseteq K\\subseteq E$ , and whose morphisms are field homomorphisms which fix $F$ pointwise (but need not commute with the inclusions into $E$ ).There is another strict category whose objects are subgroups $H\\subseteq G$ , and whose morphisms are morphisms of $G$ -sets $G/H\\to G/K$ .The fundamental theorem of Galois theorysays that these two precategories are isomorphic (not merely equivalent).

单词	96StrictCategories
释义	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 $A$ be a precategory and $x : A$ an object.Then there is a precategory $\\mathsf{mono}(A,x)$ as follows: • Its objects consist of an object $y : A$ and a monomorphism $m:\\hom_{A}(y,x)$ .(As usual, $m:\\hom_{A}(y,x)$ is a monomorphism (or is monic) if $(m\\circ f=m\\circ g)\\Rightarrow(f=g)$ .) • Its morphisms from $(y,m)$ to $(z,n)$ are arbitrary morphisms from $y$ to $z$ in $A$ (not necessarily respecting $m$ and $n$ ). An equality $(y,m)=(z,n)$ of objects in $\\mathsf{mono}(A,x)$ consists of an equality $p:y=z$ and an equality ${p}_{*}\\mathopen{}\\left({m}\\right)\\mathclose{}=n$ , which by \\autorefct:idtoiso-trans is equivalently an equality $m=n\\circ\\mathsf{idtoiso}(p)$ .Since hom-sets are sets, the type of such equalities is a mere proposition.But since $m$ and $n$ are monomorphisms, the type of morphisms $f$ such that $m=n\\circ f$ is also a mere proposition.Thus, if $A$ is a category, then $(y,m)=(z,n)$ is a mere proposition, and hence $\\mathsf{mono}(A,x)$ 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 $E/F$ be a finite Galois extensionof fields, and $G$ its Galois group.Then there is a strict category whose objects are intermediate fields $F\\subseteq K\\subseteq E$ , and whose morphisms are field homomorphisms which fix $F$ pointwise (but need not commute with the inclusions into $E$ ).There is another strict category whose objects are subgroups $H\\subseteq G$ , and whose morphisms are morphisms of $G$ -sets $G/H\\to G/K$ .The fundamental theorem of Galois theorysays that these two precategories are isomorphic (not merely equivalent).
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