Let $\mathcal{C}$ be a category, let $B$ be an object in $\mathcal{C}$, and let $\mathcal{R}$ be a *span* from $B$ to itself. (That is: $\mathcal{R}$ is a diagram $B \stackrel{r_1}{\longleftarrow} R \stackrel{r_2}{\longrightarrow} B$, where $R$ is some other object in $\mathcal{C}$, and $r_1$ and $r_2$ are $\mathcal{C}$-morphisms.) It is common to think of $\mathcal{R}$ as a sort of "abstract binary relation" on $B$. (Indeed, if $\mathcal{C}$ is the category of sets, then it is easy to represent every "ordinary" binary relation on a set $B$ as a span, and to interpret every span on $B$ as an ordinary binary relation.)

If $B$ is a set, then a *weak order* on $B$ is a binary relation on $B$ which is *complete*, *reflexive*, and *transitive*. I am interested in defining something analogous to a weak order in an abstract category, so I need something analogous to these three properties for abstract spans.

Reflexivity is straightforward. If $\mathcal{R}$ is a span from $B$ to itself an abstract category, then we can define $\mathcal{R}$ to be *reflexive* if it extends the identity span (that is, the span $B \stackrel{I}{\longleftarrow} B \stackrel{I}{\longrightarrow} B$, where $I$ is the identity morphism). But the definition for the other two properties is not so obvious. So my question is this:

Is there a way to reformulate the properties of

completenessandtransitivityfor spans on abstract categories?

(To be clear, a binary relation $\succeq$ on a set $B$ is *complete* (or *total*) if, for all $a,b\in B$, either $a\succeq b$ or $b\succeq a$. The binary relation $\succeq$ is *transitive* if, for all $a,b,c\in B$, if $a\succeq b$ and $b\succeq c$, then $a\succeq c$.)

If $\mathcal{C}$ admits all pullbacks, then we can define the "composition" of the span $\mathcal{R}$ with itself in the obvious way, to obtain a span that I will denote by $\mathcal{R}^2 = (B \stackrel{q_1}{\longleftarrow} Q \stackrel{q_2}{\longrightarrow} B)$, for some object $Q$ and morphisms $q_1$ and $q_2$. Then we can define $\mathcal{R}$ to be *transitive* if $\mathcal{R}^2$ is "extended" by $\mathcal{R}$, by which I mean there is a commuting diagram

$$ \begin{array}{ccccc} & & Q \\ &\stackrel{q_1}{\swarrow} & \downarrow & \stackrel{q_2}{\searrow} \\ B &\stackrel{r_1}{\leftarrow} & R & \stackrel{r_2}{\rightarrow} & B \end{array} $$

However, if $\mathcal{C}$ does *not* admit all pullbacks, then this strategy doesn't work; is there another way to define "transitive" in this context?

This seems like an obvious question, and I would not be surprised if someone already answered it years ago. However, I am not an expert in category theory, and I have not been able to find an answer in any of the obvious places. I would be very grateful if someone could point me to any literature or make any other suggestions.

Dedekind-completeness---i.e. the existence of suprema and infima for arbitrary subsets.) However, I wouldavoidthe word "linear", because alinearorder is alsoantisymmetric---i.e. it has the property that $(a \succeq b$ and $b\succeq a)$ if and only if $a=b$. But let's not get bogged down in terminology issues. $\endgroup$ – Marcus Pivato Jul 5 '16 at 7:20