TY - JOUR

T1 - Category theoretic structure of setoids

AU - Kinoshita, Yoshiki

AU - Power, John

PY - 2014/8

Y1 - 2014/8

N2 - A setoid is a set together with a constructive representation of an equivalence relation on it. Here, we give category theoretic support to the notion. We first define a category Setoid and prove it is cartesian closed with coproducts. We then enrich it in the cartesian closed category Equiv of sets and classical equivalence relations, extend the above results, and prove that Setoid as an Equiv-enriched category has a relaxed form of equalisers. We then recall the definition of E-category, generalising that of Equiv-enriched category, and show that Setoid as an E-category has a relaxed form of coequalisers. In doing all this, we carefully compare our category theoretic constructs with Agda code for type-theoretic constructs on setoids.

AB - A setoid is a set together with a constructive representation of an equivalence relation on it. Here, we give category theoretic support to the notion. We first define a category Setoid and prove it is cartesian closed with coproducts. We then enrich it in the cartesian closed category Equiv of sets and classical equivalence relations, extend the above results, and prove that Setoid as an Equiv-enriched category has a relaxed form of equalisers. We then recall the definition of E-category, generalising that of Equiv-enriched category, and show that Setoid as an E-category has a relaxed form of coequalisers. In doing all this, we carefully compare our category theoretic constructs with Agda code for type-theoretic constructs on setoids.

KW - setoid

KW - proof assistant

KW - proof irrelevance

KW - Cartesian closed category

KW - coproduct

KW - Equiv-category

KW - Equiv-inserter

KW - E-category

KW - E-coinserter

UR - http://dx.doi.org/10.1016/j.tcs.2014.03.006

U2 - 10.1016/j.tcs.2014.03.006

DO - 10.1016/j.tcs.2014.03.006

M3 - Article

VL - 546

SP - 145

EP - 163

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -