nLab
partially ordered dagger-category

Definition

A locally partially ordered category is a category together with partial order $\subseteq$ on each of its hom-sets such that

$$f_1\subseteq f_2\wedge g_1\subseteq g_2\Rightarrow g_1\circ f_1\subseteq g_2\circ f_2$$f_1 \subseteq f_2 \;\wedge\; g_1 \subseteq g_2 \;\Rightarrow\; g_1 \circ f_1 \subseteq g_2 \circ f_2

for any morphisms $f_1$, $g_1$, $f_2$, $g_2$ such that the composites above are defined. In other words, a category enriched in Pos, the category of posets.

Compare this to the notion of category internal to Pos.

Similarly, such a partially ordered category can be considered as a special kind of double category, while a locally partially ordered category can be considered as a special kind of $2$-category.

Old discussion:

Sridhar Ramesh: Is there meant to also be a partial ordering on the objects in addition to those on the Hom-sets? (Without this, I cannot make sense of the source and target maps preserving the partial order. Indeed, as it stands, I don’t see how this definition is any different from that of a locally partially ordered category.)

David Roberts: I agree: the first definition is of a category enriched in Pos, whereas the reference to source and target maps clearly talks about an internal category. I’ve edited it.

Partially ordered $†$-categories

For a partially ordered $†$-category we can (optionally?) require that

(1)$$f^{†}\subseteq g^{†}\iff f\subseteq g$$f^{\dagger} \subseteq g^{\dagger} \Leftrightarrow f \subseteq g

for any morphisms $f$ and $g$.

The following nomenclature points were raised by one of the previous contruibutors. What do people think? Are they justified or in conflict with existing terminology? Are there examples where the conditions come up naturally?

• For a partially ordered $†$-category I will call monovalued morphism such a morphism $f$ that $f\circ f^{†}\subseteq 1_{\mathrm{Dst}f}$.

• For a partially ordered category with inverses I will call entirely defined morphism such a morphism $f$ that $f^{†}\circ f\supseteq 1_{\mathrm{Src}f}$.

Partially ordered groupoids

The particular case of an ordered groupoid (in which each $f^{-1}$ is the inverse of $f$) is called an ordered groupoid. This has been studied extensively by Mark Lawson, for instance see

• Lawson, Mark V Constructing ordered groupoids Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 2 (2005), p. 123-138, (numdam)

or his book:

• M.V. Lawson, Inverse semigroups: the theory of partial symmetries , World Scientific, 1998.

He makes the point that they were initially studied by Ehresmann and are very closely related to inverse semi-groups.

See also

• generalized continuity