nLab pseudo-proset

Contents

Contents

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 AA is a set EE of morphisms and functions s:EVs:E \to V, t:EVt:E \to V (a loop directed pseudograph), with functions id:VE\mathrm{id}:V \to E and

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

such that

  • for every aVa \in V, s(id(a))= Eas(\mathrm{id}(a)) =_E a
  • for every aVa \in V, t(id(a))= Eat(\mathrm{id}(a)) =_E a
  • for every fEf \in E and gEg \in E such that t(f)= Vs(g)t(f) =_V s(g), s(comp(f,g))= Es(f)s(\mathrm{comp}(f,g)) =_E s(f)
  • for every fEf \in E and gEg \in E such that t(f)= Vs(g)t(f) =_V s(g), t(comp(f,g))= Et(g)t(\mathrm{comp}(f,g)) =_E t(g)

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

Examples

A category is a pseudo-proset AA which additionally satisfies

  • for every object aOb(A)a \in Ob(A), bOb(A)b \in Ob(A), cOb(A)c \in Ob(A), and dOb(A)d \in Ob(A) and edge fMor A(a,b)f \in Mor_A(a, b), gMor A(b,c)g \in Mor_A(b, c), and hMor A(c,d)h \in Mor_A(c, d)

    comp(a,b,d)(f,comp(b,c,d)(g,h))=comp(a,c,d)(comp(a,b,c)(f,g),h)\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 aOb(A)a \in Ob(A), bOb(A)b \in Ob(A) and edge fMor A(a,b)f \in Mor_A(a, b)

    comp(a,b,b)(f,id(b))=f\mathrm{comp}(a, b, b)(f, \mathrm{id}(b)) = f
  • for every vertex aOb(A)a \in Ob(A), bOb(A)b \in Ob(A) and edge fMor A(a,b)f \in Mor_A(a, b)

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

See also

Last revised on September 22, 2022 at 11:16:14. See the history of this page for a list of all contributions to it.