nLab setoid

Idea

A setoid is a set equipped with a pseudo-equivalence relation. In the same way that magmoids are the raw structure used to build semicategories and pseudo-prosets are the raw structure used to build categories, setoids are the raw structure used to build dagger categories and groupoids (i.e. a groupoid without associativity, unital laws, and inverse laws). Setoids are also sometimes used in “impoverished” foundations of mathematics that lack a primitive notion of quotient set; see for instance Bishop set.

Note on terminology

Sometimes, the term “setoid” is used in the mathematics literature for a set equipped with an equivalence relation, rather than a set equipped with a pseudo-equivalence relation. However, in the nLab we shall by default take setoids as those sets equipped with a pseudo-equivalence relation, which is required for the category $Setoid$ to be the ex/lex completion of $Set$.

Definition

With a set and a family of sets

Using graph theoretic terminology, a setoid is a loop directed pseudograph, a set $V$ of vertices with a family of sets $E(a, b)$ of edges for each vertex $a \in A$ and $b \in A$, which comes with the following additional structure

• for each vertex $a \in A$ a family of edges

  $$\mathrm{refl}(a) \in E(a, a)$$

• for each vertex $a \in A$ and $b \in A$ a family of functions

  $$\mathrm{sym}(a, b):E(a, b) \to E(b, a)$$

• for each vertex $a \in A$, $b \in A$, and $c \in A$, a family of functions

  $$\mathrm{trans}(a, b, c):(E(a, b) \times E(b, c)) \to E(a, c)$$

The edge family $(E, \mathrm{refl}, \mathrm{sym}, \mathrm{trans})$ is called a pseudo-equivalence relation.

With two sets

A setoid is a set $V$ with a set $E$ and functions $s:E \to V$, $t:E \to V$ (a loop directed pseudograph), with with functions $\mathrm{refl}:V \to E$, $\mathrm{sym}:E \to E$, and

$$\mathrm{trans}:\{(f,g) \in E \times E \vert t(f) =_V s(g)\} \to E$$

such that

• for every $a \in V$, $s(\mathrm{refl}(a)) =_E a$
• for every $a \in V$, $t(\mathrm{refl}(a)) =_E a$
• for every $f \in E$, $s(f) =_V t(\mathrm{sym}(f))$
• for every $f \in E$, $t(f) =_V s(\mathrm{sym}(f))$
• for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $s(\mathrm{trans}(f,g)) =_E s(f)$
• for every $f \in E$ and $g \in E$ such that $t(f) =_V s(g)$, $t(\mathrm{trans}(f,g)) =_E t(g)$

The structure $(E, s, t, \mathrm{refl}, \mathrm{sym}, \mathrm{trans})$ is called a pseudo-equivalence relation.

Extensional functions, dagger functors, and isomorphisms

We define an extensional function $f:A \to B$ between two setoids $A = (V_A, E_A, \mathrm{refl}_A, \mathrm{sym}_A, \mathrm{trans}_A)$ and $B = (V_B, E_B, \mathrm{refl}_B, \mathrm{sym}_B, \mathrm{trans}_B)$ to be a function $f_V:V_A \to V_B$ and a family of functions $f_E(a, b):E_A(a, b) \to E_B(f_V(a), f_V(b))$

Composition of extensional functions is defined as composition of the vertex function and of each edge function.

An extensional function between two setoids is full if every edge function $f_E(a, b):E_A(a, b) \to E_B(f_V(a), f_V(b))$ is a surjection, and an extensional function is faithful if every edge function is an injection. An extensional function is injective-on-objects? if the vertex function $f_V:V_A \to V_B$ is an injection, surjective-on-objects if the vertex function is a surjection, and bijective-on-objects if the vertex function is a bijection.

An extensional function is a dagger functor if additionally it preserves the functions $\mathrm{refl}$, $\mathrm{sym}$, $\mathrm{trans}$

$$f_E(a, a)(\mathrm{refl}_A(a)) = \mathrm{refl}_B(f_V(a))$$

$$f_E(a, b)(\mathrm{sym}_A(a, b)(f)) = \mathrm{sym}_B(f_V(a), f_V(b))(f_E(a, b)(f))$$

$$f_E(a, c)(\mathrm{trans}_A(f,g)) = \mathrm{trans}_B(f_E(a, b)(f), f_E(b, c)(g)$$

An isomorphism of setoids is a full and faithful bijective-on-objects dagger functor.

Equivalence of definitions

Given a one-set-of-edges setoid $E\rightrightarrows V$, we define a family-of-sets-of-edges setoid by taking $E(x,y)$ to be the preimage of $(x,y)$ under the function $(s,t):E \to V\times V$. Conversely, given a family-of-sets-of-edges setoid we define a one-set-of-edges setoid by taking $E$ to be the disjoint union of the families of edges $E = \coprod_{x,y\in V} E(x,y)$.

…

Examples

Every category with a contravariant endofunctor that is the identity-on-objects is a setoid where

• for every vertex $a \in V$, $b \in V$, $c \in V$, and $d \in V$ and edge $f \in E(a, b)$, $g \in E(b, c)$, and $h \in E(c, d)$

  $$\mathrm{trans}(a, b, d)(f, \mathrm{trans}(b, c, d)(g, h)) = \mathrm{trans}(a, c, d)(\mathrm{trans}(a, b, c)(f, g), h)$$

• for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$

  $$\mathrm{trans}(a, b, b)(f, \mathrm{refl}(b)) = f$$

• for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$

  $$\mathrm{trans}(a, a, b)(\mathrm{refl}(a), f) = f$$

This includes dagger categories, where additionally

• for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$

  $$\mathrm{sym}(a, b)(\mathrm{sym}(b, a)(f)) = f$$

• for every vertex $a \in V$, $b \in V$, $c \in V$ and edge $f \in E(a, b)$, $g \in E(b, c)$

  $$\mathrm{sym}(a, c)(\mathrm{trans}(a, b, c)(f, g)) = \mathrm{trans}(c, b, a)(\mathrm{sym}(a, b)(f), \mathrm{sym}(b, c)(g))$$

• for every vertex $a \in V$

  $$\mathrm{sym}(a, a)(\mathrm{refl}(a)) = \mathrm{refl}(a)$$

and groupoids, where

• for every vertex $a \in V$, $b \in V$ and edge $f \in E(a, b)$

  $$\mathrm{trans}(a, b, a)(f, \mathrm{sym}(a, b)(f)) = \mathrm{refl}(a)$$

• for every vertex $a \in V$, $b \in V$ and edge $f:E(a, b)$

  $$\mathrm{trans}(b, a, b)(\mathrm{sym}(a, b)(f), f) = \mathrm{refl}(b)$$

Additionally, a setoid with one vertex is equivalent to a pointed magma with an endofunction.

Thin setoids

Recall in graph theory that a loop directed pseudograph is simple or a loop digraph if for every vertex $a \in V$ and $b \in V$, the set of edges $E(a, b)$ is a subsingleton: for every edge $f \in E(a, b)$ and $g \in E(a, b)$, $f = g$. In the other definition, a loop directed pseudograph is a loop digraph if the functions $s:E \to V$ and $t:E \to V$ are jointly monic.

A setoid is thin or simple if its underlying loop directed pseudograph is a loop digraph. In both cases, the pseudo-equivalence relation becomes an equivalence relation. The term “thin” originates from category theory, while the term “simple” originates from graph theory.

A thin setoid is equivalently a thin dagger category, or a dagger category enriched in truth values. A thin setoid is also a thin groupoid, or a groupoid enriched in truth values.

Sometimes in the mathematical literature, setoids are thin by default.

Core of a setoid

For any setoid $A$, the core of $A$ is defined as the maximal subgroupoid? $\mathrm{Core}(A)$ of $A$.

More specifically, a subsetoid $G$ of a setoid $A$ is a setoid $G$ with an faithful injective-on-objects dagger functor $f:G \to A$. A subsetoid $G$ of $A$ is a subgroupoid if the pseudo-equivalence relation of $G$ additionally satisfy the groupoid equational axioms:

• for every vertex $a \in V_G$, $b \in V_G$, $c \in V_G$, and $d \in V_G$ and edge $f \in E_G(a, b)$, $g \in E_G(b, c)$, and $h \in E_G(c, d)$

  $$\mathrm{trans}(a, b, d)(f, \mathrm{trans}(b, c, d)(g, h)) = \mathrm{trans}(a, c, d)(\mathrm{trans}(a, b, c)(f, g), h)$$

• for every vertex $a \in V_G$, $b \in V_G$ and edge $f \in E_G(a, b)$

  $$\mathrm{trans}(a, b, b)(f, \mathrm{refl}(b)) = f$$

• for every vertex $a \in V_G$, $b \in V_G$ and edge $f \in E_G(a, b)$

  $$\mathrm{trans}(a, a, b)(\mathrm{refl}(a), f) = f$$

• for every vertex $a \in V_G$, $b \in V_G$ and edge $f \in E_G(a, b)$

  $$\mathrm{trans}(a, b, a)(f, \mathrm{sym}(a, b)(f)) = \mathrm{refl}(a)$$

• for every vertex $a \in V_G$, $b \in V_G$ and edge $f:E_G(a, b)$

  $$\mathrm{trans}(b, a, b)(\mathrm{sym}(a, b)(f), f) = \mathrm{refl}(b)$$

A subgroupoid $G$ of a setoid $A$ is a maximal subgroupoid if the dagger functor $f:G \to A$ is bijective-on-objects, and additionally if, for every other subgroupoid $H$ of $A$ with faithful injective-on-objects dagger functor $g:H \to A$, there is a unique faithful injective-on-objects dagger functor $h:H \to G$ such that $h \circ f = g$.

Category of setoids

Let the category $Set$ be defined as a category that is finitely complete and well-pointed (i.e. whose terminal object is a extremal generating object). This category of sets is not even a regular category, let alone an exact category, as can happen in certain foundations of mathematics, such as bare set-level Martin-Löf type theory, or ZFC and ETCS without the powerset axiom. The category $Setoid$ of setoids is then the ex/lex completion of Set.

More specifically, using the definition of setoid with two sets: if $(s_R, t_R):R\rightrightarrows X$ and $(s_S, t_S):S\rightrightarrows Y$ are two setoids, a morphism between them in $Setoid$ is defined to be a function $f:X\to Y$ with functions $f_1:R \to S$ with $s_S \circ f_1 = f \circ s_R$ and $t_S \circ f_1 = f \circ t_R$. Moreover, we declare two such functions $f,g:X\to Y$ to be equal if there exists a function $h:X\to S$ such that $s \circ h = f$ and $t \circ h = g$. Because $(s_S, t_S):S\rightrightarrows Y$ is a pseudo-equivalence relation, this defines an actual equivalence relation on the morphisms $f:X\to Y$, which is compatible with composition; thus we have a well-defined category $Setoid$ of setoids, which is the ex/lex completion of $Set$.

We have a full and faithful functor $Set\to Setoid$ sending an object $X$ to the pseudo-equivalence relation $(s_X, t_X):X\rightrightarrows X$. One can then verify directly that $Setoid$ is exact, that this embedding preserves finite limits, and that it is universal with respect to lex functors from $Set$ into exact categories.

If the presentation axiom (a weak form of the full axiom of choice) holds in $Set$, as a subcategory of $Setoid$, $Set$ could be thought of as the category of completely presented sets, the category of sets with a projective presentation. If the axiom of choice holds in $Set$, then $Set$ is equivalent to $Setoid$, as the axiom of choice implies that $Set$ is its own free exact completion, and $Set$ is equivalent to $Setoid$ because the free functor from $Set$ to $Setoid$ is an equivalence of categories.

This construction could be generalized to any finitely complete category $C$, from which the category of setoid objects of $C$ is the ex/lex completion of $C$, and denoted as $C_{ex/lex}$. If every epimorphism in $C$ is additionally a split epimorphism, then $C$ is its own free exact completion.

In type theory

Many set-level type theories use types which are 0-truncated and thus h-sets by default because of the inclusion of uniqueness of identity proofs or axiom K to the rule system of the type theory. Sometimes, these type theories do not have quotients and image factorizations, and as a result, setoids are used instead of the native h-sets.

In homotopy type theory, and more generally in any intensional type theory where not all types are h-sets, such as in a type theory without uniqueness of identity proofs or axiom K, the notion of “setoid” bifurcates into multiple distinct notions. In analogy with category, the definitions used above in this article could be called a strict setoid. When the vertex types are only required to be a type rather than a set, then this defines a presetoid. There is also the notion of a univalent setoid, where equality of vertices is isomorphism of vertices in the core of a presetoid.

Applications

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.

Fixing inadequate foundations

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+$\epsilon$.

See also

• Bishop set, quotient type

• directed graph

• pseudo-proset

• equivalence relation

• ex/lex completion

• extensional function

• dagger functor

• presetoid

• univalent setoid

• setoid object

• completely presented set

algebraic structure	oidification
truth value	preorder
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

References

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

• Martin Hofmann, §1.3, §5.1 in: Extensional concepts in intensional type theory, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [ECS-LFCS-95-327, pdf, doi:10.1007/978-1-4471-0963-1]

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]