nLab pseudo-equivalence relation

category theory

In higher category theory

Graph theory

graph theory

graph

category of simple graphs

Contents

Idea

A pseudo-equivalence relation is like an equivalence relation, but where we allow more than one element in the relation; i.e. the underlying directed graph of the relation is a directed pseudograph. In the same way that magmoids are the raw structure used to build semicategories and categories, sets with pseudo-equivalence relations are the raw structure used to build dagger categories and groupoids (i.e. a groupoid without associativity, unital laws, and inverse laws).

Sets equipped with pseudo-equivalence relations are sometimes called setoids (i.e. in Wilander (2012), §4; Palmgren & Wilander (2014), §2, §6; Emmenegger & Palmgren (2020), §6; Cipriano (2020), Def. 1.1.1), but the term “setoid” is also used in mathematics to refer only to the sets with equivalence relations. In this article, the term “setoid” is used in the former sense of “sets equipped with pseudo-equivalence relations”.

Definition

With a family of sets

Using graph theoretic terminology, a pseudo-equivalence relation on a set $V$ of vertices is a family of sets $E(a, b)$ of edges for each vertex $a \in V$ and $b \in V$, which comes with the following additional structure

• for each vertex $a \in V$ an element

$\mathrm{refl}(a) \in E(a, a)$
• for each vertex $a \in V$ and $b \in V$ a family of functions

$\mathrm{sym}(a, b):E(a, b) \to E(b, a)$
• for each vertex $a \in V$, $b \in V$, and $c \inV$, a family of functions

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

With a set

A pseuodo-equivalence relation on a set $V$ is 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 $t(f) =_V s(g)$, $t(\mathrm{trans}(f,g)) =_E t(g)$

Extensional functions, dagger functors, and isomorphisms

We define an extensional function $f:A \to B$ between two sets $A$ and $B$ with pseudo-equivalence relations $(E_A, \mathrm{refl}_A, \mathrm{sym}_A, \mathrm{trans}_A)$ and $(E_B, \mathrm{refl}_B, \mathrm{sym}_B, \mathrm{trans}_B)$ to be a function $f_V:A \to B$ 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 sets with pseudo-equivalence relations 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

The morphisms $E(a, b)$ of a category $V$ with a contravariant endofunctor that is the identity-on-objects is a pseudo-equivalence relation 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 pseudo-equivalence relation on a set 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 pseudo-equivalence relation

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

More specifically, a sub-pseudo-equivalence relation of a pseudo-equivalence relation $A$ is a set $G$ with a pseudo-equivalence relation and an faithful injective-on-objects dagger functor $f:G \to A$. A sub-pseudo-equivalence relation $G$ of $A$ is a subgroupoid if $G$ 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 $f:E_G(a, b)$

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

Every pseudo-equivalence relation has at least one subgroupoid given by only the identity morphisms $\mathrm{refl}(a):E_A(a, a)$ of $A$. A subgroupoid $G$ of a pseudo-equivalence relation $A$ 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 pseudo-equivalence relations

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 $\mathrm{PseudoEquivRel}$ of sets with pseudo-equivalence relations is then the ex/lex completion of Set.

More specifically, using the definition of pseudo-equivalence relation with two sets: if $(s_R, t_R):R\rightrightarrows X$ and $(s_S, t_S):S\rightrightarrows Y$ are two pseudo-equivalence relations, a morphism between them in $\mathrm{PseudoEquivRel}$ 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 $\mathrm{PseudoEquivRel}$ of pseudo-equivalence relations, which is the ex/lex completion of $Set$.

We have a full and faithful functor $Set\to \mathrm{PseudoEquivRel}$ sending an object $X$ to the pseudo-equivalence relation $(s_X, t_X):X\rightrightarrows X$. One can then verify directly that $\mathrm{PseudoEquivRel}$ 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 $\mathrm{PseudoEquivRel}$, $Set$ 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 $\mathrm{PseudoEquivRel}$, as the axiom of choice implies that $Set$ is its own free exact completion, and $Set$ is equivalent to $\mathrm{PseudoEquivRel}$ because the free functor from $Set$ to $\mathrm{PseudoEquivRel}$ is an equivalence of categories.

This construction could be generalized to any finitely complete category $C$, from which the category of internal pseudo-equivalence relations 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.

References

For setoids defined as a set with an pseudo-equivalence relation:

Last revised on July 7, 2023 at 18:15:07. See the history of this page for a list of all contributions to it.