nLab enriched preorder

Contents

Contents

Idea

Recall (here) that a preorder may be understood as a thin (0,1)-category, hence as a thin category enriched over the cartesian monoidal proset of truth values. In generalization, one may speak of enriching preorders over other monoidal posets.

Defintion

Let (M,,,)(M, \leq, \wedge, \top) be a monoidal poset. A MM-enriched proset or proset enriched over/in MM is a set PP with a binary function o:P×PMo:P \times P \to M such that

  • for every aPa \in P, bPb \in P, and cPc \in P, o(a,b)o(b,c)o(a,c)o(a, b) \wedge o(b, c) \leq o(a, c)

  • for every aPa \in P, o(a,a)\top \leq o(a, a).

Examples

See also

Last revised on September 22, 2022 at 14:40:57. See the history of this page for a list of all contributions to it.