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Sorry for a duplicated post, but duplicated research would be even worse.
From this USENET post:
Has anyone heard about categories with ordered set of objects and embeddings $\operatorname{Hom}(A_0,B_0) \rightarrow Hom(A_1,B_1)$ when $A_0\leq A_1$ and $B_0\leq B_1$?
That is it is specified: 1. a category; 2. an order (or preorder?) on objects; 3. aforementioned embeddings (by an embedding I mean an injective function preserving composition).
One example: Reloid is a triple $(A;B;F)$ where $A$ and $B$ are sets and $F$ is a filter on the cartesian product $A\times B$. It turns to be a category when we define composition $G\circ F$ as the filter corresponding to the base $\{ g\circ f | f\in F, g\in G \}$.
When $A_0\subseteq A_1$ and $B_0\subseteq B_1$ we have $A_0\times B_0\subseteq A_1\times B_1$ and thus there is an obvious embedding of filters on $A_0\times B_0$ into filters on $A_1\times B_1$.
From my drafts:
Definition. A hierarchical category is a category with a partially ordered [TODO: or preordered?] set of objects and an injective function preserving composition $\operatorname{Hom} ( A_0 ; A_1) \rightarrow \operatorname{Hom} ( B_0 ; B_1)$ defined whenever $A_0 \leq B_0$ and $A_1 \leq B_1$.
Definition. A category with embeddings of objects is a category with a partially ordered [TODO: or preordered?] set of objects and morphism $A \hookrightarrow B : A \rightarrow B$ (embedding of $A$ into $B$) defined for every objects $A \leq B$.
Definition. A dagger category with embeddings of objects is a category which is both a dagger category and a category with embeddings of objects.
Obvious. Every dagger category with embeddings of objects induces a hierarchical category by the formula $f \mapsto ( A_1 \hookrightarrow B_1) \circ f \circ ( A_0 \hookrightarrow B_0)^{\dagger}$ for $f : A_0 \rightarrow A_1$ and $A_0 \leq B_0$ and $A_1 \leq B_1$.
In other words, a category with embeddings of objects is a category with a partially ordered [or preordered?] set of objects and a functor from the poset of objects into the category.
I'd say the same thing as in #3, and partial order or preorder makes no difference up to equivalence (assuming the axiom of choice, or alternatively using anafunctors).
But I'm not sure that this is what you want! The sets $A$ and $B$ in a reloid should only be abstract sets; that is, even if you are working in $ZFC$ (or something similar), you don't really care about the membership tree of its elements (and elements' elements, etc). And indeed (if I'm not confused), you not only have an injection $\operatorname{Reld}(A_0,B_0) \rightarrowtail \operatorname{Reld}(A_1,B_1)$ when $A_0 \subseteq A_1$ and $B_0 \subseteq B_1$; you have such an injection for every pair of injections $A_0 \rightarrowtail A_1$ and $B_0 \rightarrowtail B_1$.
So I think that you're looking for a double category, which has two kinds of morphisms; in this case, one kind is the injections and one kind is the reloids. (Although, you might consider whether you can generalize further from injections to arbitrary functions.)
I can’t open your link (nLab wiki does not work again).
What you suggest me is generalization for the sake of generalization. It is not always good. I see no any reason whatsoever to consider arbitrary injections instead of set embeddings only.
The notion of subsets is used by me to define “restriction” of a funcoid or reloid to this set (just like to restriction of a topological space to a set). This is in turn to be used to define equalizers and co-equalizers of funcoids or reloids. (I try to calculate equalizers to prove the theorems that in my categories all (co)products exist.)
@porton Toby’s suggestion is not “generalization for the sake of generalization” (and btw, that sounds pretty rude to me). It’s more that he’s ’generalizing’ so that the concept becomes invariant under categorical equivalence. Experience has shown that this is generally a wise move.
Whether equalizers and coequalizers exist will not depend on whether you use subsets or arbitrary injections – again by categorical equivalence.
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