Setoids in type theory

Abstract

Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach.

Triposes, exact completions, and Hilbert's ε-operator

“…Other examples come from theories used in the formalization of constructive mathematics: the category of total setoidsà la Bishop and functional relations based on the Minimalist Type Theory in [Mai09], which coincides with the exact completion Ex G mtt where the doctrine G mtt is defined as in [MR13b], or the category of total setoidsà la Bishop and functional relations based on the Calculus of Constructions [Coq90], which coincides with the exact completion Ex G CoC where the doctrine G CoC is constructed from the Calculus of Constructions as G mtt in [MR13b], and it forms a topos as mentioned in [BCP03].…”

Abstract effects and proof-relevant logical relations

“…We also omit | − | wherever appropriate. We remark that "setoids" also appear in constructive mathematics and formal proof, see e.g., [3], but the proof-relevant nature of equality proofs is not exploited there and everything is based on sets (types) rather than predomains. A morphism from setoid A to setoid B is an equivalence class of pairs …”

The grounding of computational psychoanalysis: A comparative history of culture overview of Matte Blanco bilogic

“…The groupoid structure has been proved to be at the early bases of almost all the most basic commutative and non-commutative algebraic structures, so that they lie at the deeper roots of the general algebraic formalization. In particular, groupoid structures are also at the basis of graph and combinatorial structures (see [15] and references therein) as well as having applications in type theory (see [16]); likewise, ordered groupoids are at the foundations of other algebraic structures, like groups and inverse semigroups (see [17]). Finally, groupoids have recently received remarkable attention also in non-linear dynamics of networks: in this regard, see [7], where a very interesting discussion of network synchrony, asynchrony and related symmetry breaking phenomena, in the context of groupoid formalism, is made 6 .…”

“…Only through the transference intervention of the psychoanalyst, it was possible to try to restore a stronger paternal role, so recovering the right time perception, hence stemming maternal symmetric thought drives. John gradually re-acquired his own sense of individuality and awareness, a right time perception, a normal balanced relationship between physical and psychological time 16 , all elements, these, which 14 For a technical treatment of the theory of symmetry breaking via group action, see, for example, [31] as regard graph theory, and [32] as regard bifurcation symmetry breaking theory. 15 On the symmetry or reciprocity character of double bind, as well as on other interesting remarks and considerations, see [28, 6.43] and references therein.…”