Inverse Image Category (Preimage Category)

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Definition of inverse image category
Isomorphisms of inverse image category
Extending operators onto inverse image category

This short article, where I define inverse image category by a function of any given category $C$ , is mainly terminological note to be used as a base of my other category theory articles.

Definition of inverse image category

Let $C$ (original category) is a category and $λ$ is a function whose image is a subset of the set of objects of $C$ .

I will call inverse image category (or preimage category) of $C$ by $λ$ and denote $λ^{-1} C$ the category:

whose set of objects is the domain of function $λ$ ;
whose morphisms from an object $a$ to an object $b$ are all triples $(f, a, b)$ where $f$ is a morphism of $C$ , $Src f = λ (a)$ and $Dst f = λ (b)$ ;
whose composition of morphisms $(f, a, b)$ and $(g, b, c)$ is $(g \circ f, a, c)$ .

It is really a category because for an object $a$ there exists identity morphism, namely $1_{a} = (1_{λ (a)}, a, a)$ .

After this definition $λ$ extended to the set of morphisms of the inverse image category by the formula $λ (f, a, b) = f$ becomes a functor from the preimage category to our original category.

It is really a functor because it obviously preserves composition and maps identity to identity.

We will additionally define compositions of morphisms of the inverse image category with morphisms of the original category by the formulas

$g \circ (f, a, b) = g \circ f$ ;
$(g, a, b) \circ f = g \circ f$ .

(That is in such compositions we lose the information (values of $a$ and $b$ ) about source and destination.)

Isomorphisms of inverse image category

Theorem A morphism $(f, a, b)$ is an isomorphism of the category $λ^{-1} C$ if and only if $f$ is an isomorphism of the category $C$ . If it is an isomorphism then ${(f, a, b)}^{-1} = (f^{-1}, b, a) .$

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If $f$ is an isomorphism, then

$(f^{-1}, b, a) \circ (f, a, b) = (f^{-1} \circ f, a, a) = (1_{dom f}, a, a) = 1_{a}$ ;
$(f, a, b) \circ (f^{-1}, b, a) = (f \circ f^{-1}, b, b) = (1_{im f}, b, b) = 1_{b}$ .

So $(f, a, b)$ is an isomorphism, ${(f, a, b)}^{-1} = (f^{-1}, b, a)$ .

Let now $(f, a, b)$ is an isomorphism. Then for some $f^{/}$ we have

$(f^{/}, b, a) \circ (f, a, b) = 1_{a}$ ; $(f^{/} \circ f, a, a) = (1_{dom f}, a, a)$ .

Consequently $f^{/} \circ f = 1_{dom f}$ . Analogously $f \circ f^{/} = 1_{im f}$ . So $f$ is an isomorphism.

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Extending operators onto inverse image category

For any operator $Ψ$ on the set of morphisms of $C$ , which preserves source and destination of morphisms (that is is closed on every set $Hom (A, B)$ ), this operator can be extended to the inverse image category by the formula $Ψ (f, a, b) = (Ψ (f), a, b) .$

Likewise any operator $Ψ$ which reverses the source and destination of a morphism (that is maps any set $Hom (A, B)$ to $Hom (B, A)$ ), this operator can be extended to the inverse image category by the formula $Ψ (f, a, b) = (Ψ (f), b, a) .$

The above applies also for operators partially defined on the set of morphisms of $C$ .

In these cases when for a function $Ψ$ both of the above definitions apply, by which of the two variants to extend $Ψ$ on the inverse image category should be normally clear from the context (e.g. how $Ψ$ acts on other categories).

The above definitions conform with the properties of reverse of an isomorphism $Ψ = f \mapsto f^{-1}$ (which is a partial unary operator which transposes source and destination): ${(f, a, b)}^{-1} = (f^{-1}, b, a) .$

Having a relation $\subseteq$ (e.g. a partial order) of morphisms of the category $C$ , defined at least for morphisms having the same source and destination $A$ and $B$ , we can extend it to pairs of morphisms of a preimage category having the same source and destination $a$ and $b$ such that $λ (a) = A$ and $λ (b) = B$ by the formula $(f, a, b) \subseteq (g, a, b) \Leftrightarrow f \subseteq g .$

Note that this maps a partial order on $Hom (A, B)$ to a partial order on $Hom (a, b)$ . So essentially inverse image maps a partially ordered category to a partially ordered category. (The partial order on the preimage category can be taken for example as the union of all partial orders on the sets $Hom (a, b)$ (where $a$ and $b$ are objects of the preimage category).

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