Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
basic constructions:
strong axioms
further
In constructive mathematics, such as in intuitionistic type theory, the notion of setoids [Hofmann (1995)] is a formalization of the idea of Bishop sets (following Bishop (1967)) and serves as a way of “constructively” speaking about quotient sets without, in a sense, actually constructing these: A setoid is essentially a set equipped with an equivalence relation or a similar structure (such as a pseudo-equivalence relation), and thought of as a stand-in for the would-be quotient set (quotient type) by that relation.
Accounts which take setoids to be…
…actual equivalence relations, i.e. $Setoids \coloneqq (X \colon Set) \times (R \colon X \times X \to {\color{purple}Prop}) \times ReflSymTrans(R)$:
Hofmann (1995), §5.1.1; Barthe, Capretta & Pons (2003), §2.1; Stacks Project; hackage.haskell; Wikipedia;
…pseudo-equivalence relations, i.e. $Setoids \coloneqq (X \colon Set) \times (R \colon X \times X \to {\color{purple}Set}) \times ReflSymTrans(R)$:
Wilander (2012), §4; Palmgren & Wilander (2014), §2, §6; Emmenegger & Palmgren (2020), §6; Cipriano (2020), Def. 1.1.1,
… either:
Here the definition of setoids in terms of equivalence relations may be closer to the original intention of Bishop (1967) (though this remains a little vague) and is natural from the point of view of category theory/homotopy theory where setoids in this sense, and with equivalence of elements regarded as isomorphism, are exactly those groupoids which are equivalent to sets (the 0-truncated groupoids).
Another perspective is that setoids in the sense of equivalence relations consitute the ex/reg completion of Sets, while those in terms of pseudo-equivalence relations constitute the ex/lex completion of Sets.
In terms of type theory, the definition in terms of equivalence relations is natural under the propositions as some types-paradigm, while the definition in terms of pseudo-equivalence relations is natural under the types as propositions-paradigm.
In any case, beware that terminology and definitions may differ significantly across authors, cf. Palmgren (2005) p. 9.
Setoids are also sometimes used in “impoverished” foundations of mathematics that lack a primitive notion of quotient set; see for instance Bishop set.
In constructive mathematics, we want the real numbers to form a linearly ordered Heyting field $R$ with completeness for located Dedekind cuts. Using power sets and a set $N$ of natural numbers, one may form $R$ directly as a subset of $\mathcal{P}N$ (or something equivalent), but what if you wish to be (at least weakly) predicative? Using function sets, one may form the Cauchy reals as a subquotient of $N^N$, but these are complete in the desired sense only if a weak form of countable choice (which follows from either the presentation axiom or excluded middle) holds. (Essentially, there may not be enough sequences of natural numbers.) Alternatively, if you have them, you can use prefunction sets and form $R$ as a subquotient of the set of presequences of natural numbers.
The construction of $R$ above may also be done with entire relations if the axiom of fullness holds (see also real numbers object). Conversely, the axiom of fullness follows from the existence of sets of prefunctions; in addition to defining a functional entire prerelation, a prefunction between sets also defines an entire relation, and the set of these satisfies fullness. (This is related to the idea that prefunctions between sets may be formalised as entire relations.)
See also the discussion at net about how to force the domain of a net to be partial order, by using either entire relations or prefunctions as nets.
Sometimes one finds a foundation of predicative mathematics in which it appears to be impossible to construct quotient sets, leading to much hand-wringing. (For a simple example, simply remove the axiom of power sets from ZFC as normally presented.)
Assuming (as usual) that the original foundation had equality relations on its sets, there will be identity prerelations on the presets, leading to a special class of sets which we may again call the completely presented sets.
When you do this, the new kind of set is called a setoid, and then there may be hand-wringing about the need to use setoids instead of sets as one would like. But if you didn't have quotient sets originally, then you shouldn't have been talking about ‘sets’ in the first place; theories of sets without quotient sets are really theories of presets.
Some foundations also adopt an axiom of choice for functions which do not preserve the equivalence relation that, together with the identity relations, proves the presentation axiom for general sets, which means that free sets are completely presented sets.
If you are willing to accept the presentation axiom, then you can define a notion of setoid internal to a given theory of sets: as a projective set. (With the full axiom of choice, therefore, a setoid is simply a set.) Alternatively, you might forgo setoid as such but define a prefunction between sets to be an entire relation.
A similar result holds for SEAR+$\epsilon$.
The notion of “Bishop sets” goes to back the definition of sets in constructive mathematics/constructive analysis according to:
Errett Bishop, Foundations of Constructive Analysis, Mcgraw-Hill (1967)
Errett Bishop, Douglas Bridges, p. 15 of: Constructive Analysis, Grundlehren der mathematischen Wissenschaften 279, Springer (1985) [doi:10.1007/978-3-642-61667-9]
The connection to dependent type theory and the term setoid is due to
Survey of further developments:
Gilles Barthe, Venanzio Capretta, Olivier Pons, Setoids in type theory, Journal of Functional Programming 13 2 (2003) 261-293 [doi:10.1017/S0956796802004501]
Erik Palmgren, Bishop’s set theory (2005) [pdf, pdf]
See also:
Olov Wilander, Constructing a small category of setoids, Mathematical Structures in Computer Science 22 1 (2012) 103-121 [doi:10.1017/S0960129511000478, pdf]
Erik Palmgren, Olov Wilander, Constructing categories and setoids of setoids in type theory, Logical Methods in Computer Science, 10 3 (2014) lmcs:964 [arXiv:1408.1364, doi:10.2168/LMCS-10(3:25)2014]
Benno van den Berg, Ieke Moerdijk, Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories, Journal of Pure and Applied Algebra 222 10 (2018) 3137-3181 [doi:10.1016/j.jpaa.2017.11.017]
Jacopo Emmenegger, Erik Palmgren, Exact completion and constructive theories of sets, J. Symb. Log. 85 2 (2020) [arXiv:1710.10685, doi:10.1017/jsl.2020.2]
Cioffo Cipriano Junior, Def. 1.1.1 in: Homotopy setoids and generalized quotient completion [pdf, pdf]
hackage.haskell, Data-Setoid.html
Stacks Project, Categories fibred in setoids [tag:04S9]
Wikipedia, Setoid
Last revised on July 7, 2023 at 18:15:36. See the history of this page for a list of all contributions to it.