nLab pseudo-proset

Idea

Pseudo-preorders are to preorders as pseudo-equivalence relations are to equivalence relations. A set with a pseudo-preorder is then a pseudo-preordered set or a pseudo-proset.

Pseudo-prosets are important because they are the basic structure used to build many other structures like categories and setoids.

Definition

With a family of sets

…

With one set

A pseudo-preorder on a set $A$ is a set $E$ of morphisms and functions $s:E \to V$, $t:E \to V$ (a loop directed pseudograph), with functions $\mathrm{id}:V \to E$ and

$$\mathrm{comp}:\{(f,g) \in E \times E \vert t(f) =_V s(g)\} \to E$$

such that

• for every $a \in V$, $s(\mathrm{id}(a)) =_E a$
• for every $a \in V$, $t(\mathrm{id}(a)) =_E a$
• for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $s(\mathrm{comp}(f,g)) =_E s(f)$
• for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $t(\mathrm{comp}(f,g)) =_E t(g)$

A pseudo-preordered set or pseudo-proset $A$ consists of a set $Ob(A)$ with a pseudo-preorder $Mor_A, \mathrm{id}, \mathrm{comp}$.

Examples

A category is a pseudo-proset $A$ which additionally satisfies

• for every object $a \in Ob(A)$, $b \in Ob(A)$, $c \in Ob(A)$, and $d \in Ob(A)$ and edge $f \in Mor_A(a, b)$, $g \in Mor_A(b, c)$, and $h \in Mor_A(c, d)$

  $$\mathrm{comp}(a, b, d)(f, \mathrm{comp}(b, c, d)(g, h)) = \mathrm{comp}(a, c, d)(\mathrm{comp}(a, b, c)(f, g), h)$$

• for every vertex $a \in Ob(A)$, $b \in Ob(A)$ and edge $f \in Mor_A(a, b)$

  $$\mathrm{comp}(a, b, b)(f, \mathrm{id}(b)) = f$$

• for every vertex $a \in Ob(A)$, $b \in Ob(A)$ and edge $f \in Mor_A(a, b)$

  $$\mathrm{comp}(a, a, b)(\mathrm{id}(a), f) = f$$

See also

• preorder
• setoid
• category