nLab presetoid

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Idea

In set theory, a setoid is usually described as a set with a pseudo-equivalence relation. However, in homotopy type theory or more generally in intensional type theory, there are multiple different ways to define the concept of a “collection with a pseudo-equivalence relation”. Presetoids arise from the most general possible notion of “collection with a pseudo-equivalence relation” in homotopy type theory , in the same way that the notion of precategory and pregroupoid arise when defining categories and groupoids.

Definition

A pseudo-equivalence relation on a type $A$ is a family of sets $Hom(a, b)$ of morphisms for each element $a:A$ and $b:A$, which comes with the following additional structure

• for each element $a:A$ a family of morphisms

$\mathrm{id}(a):Hom(a, a)$
• for each object $a:A$ and $b:A$ a family of functions

$\mathrm{dagger}(a, b):Hom(a, b) \to Hom(b, a)$
• for each object $a:A$, $b:A$, and $c:A$, a family of functions

$\mathrm{comp}(a, b, c):(Hom(a, b) \times Hom(b, c)) \to Hom(a, c)$

A presetoid $A$ consists of a type $Ob(A)$ of objects and a pseudo-equivalence relation $(\mathrm{Hom}, \mathrm{id}, \mathrm{dagger}, \mathrm{comp})$.

Dagger functors

The structure-preserving maps between two presetoids are called dagger-functors. A dagger-functor $F:A \to B$ between two presetoids $A$ and $B$ consists of

• a function $f_{Ob}:\mathrm{Ob}(A) \to \mathrm{Ob}(B)$

• a family of functions $f_{hom}(a, b):\mathrm{Hom}_A(a, b) \to \mathrm{Hom}_B(f_{Ob}(a), f_{Ob}(b))$

• and identifications indicating that the identity morphism, dagger, and the composition operation are preserved:

$\mathrm{idpreserve}(a):f_{Hom}(a, a)(\mathrm{id}_A(a)) = \mathrm{id}_B(f_{Ob}(a))$
$\mathrm{daggerpreserve}(a, b, f):f_{Hom}(a, b)(\mathrm{dagger}_A(a, b)(f)) = \mathrm{dagger}_B(f_{Ob}(a), f_{Ob}(b))(f_{Hom}(a, b)(f))$
$\mathrm{comppreserve}(a, b, c, f, g):f_E(a, c)(\mathrm{comp}_A(f,g)) = \mathrm{comp}_B(f_{Hom}(a, b)(f), f_{Hom}(b, c)(g)$

An dagger functor between two presetoids is faithful if every morphism function $f_{Hom}(a, b):\mathrm{Hom}_A(a, b) \to \mathrm{Hom}_B(f_{Ob}(a), f_{Ob}(b))$ is an injection. A dagger functor is embedding-on-objects if the object function $f_V:V_A \to V_B$ is an embedding, and equivalent-on-objects if the object function is a equivalence of types.

Core of a presetoid

A subpresetoid $G$ of a presetoid $A$ is a presetoid $G$ with an faithful embedding-on-objects dagger functor $f:G \to A$. A subpresetoid $G$ of $A$ is a subpregroupoid if the pseudo-equivalence relation of $G$ additionally come with:

• for every object $a:Ob(G)$, $b:Ob(G)$, $c:Ob(G)$, and $d:Ob(G)$ and morphism $f:Hom_G(a, b)$, $g:Hom_G(b, c)$, and $h:Hom_G(c, d)$, an identification

$\mathrm{assoc}(a, b, c, d, f g, h):\mathrm{comp}(a, b, d)(f, \mathrm{comp}(b, c, d)(g, h)) = \mathrm{comp}(a, c, d)(\mathrm{comp}(a, b, c)(f, g), h)$
• for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification

$\mathrm{runital}(a, b, f):\mathrm{comp}(a, b, b)(f, \mathrm{id}(b)) = f$
• for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification

$\mathrm{lunital}(a, b, f):\mathrm{comp}(a, a, b)(\mathrm{id}(a), f) = f$
• for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification

$\mathrm{sec}(a, b, f):\mathrm{comp}(a, b, a)(f, \mathrm{dagger}(a, b)(f)) = \mathrm{id}(a)$
• for every object $a:Ob(G)$, $b:Ob(G)$ and morphism $f:Hom_G(a, b)$, an identification

$\mathrm{ret}(a, b, f):\mathrm{comp}(b, a, b)(\mathrm{dagger}(a, b)(f), f) = \mathrm{id}(b)$

A subpregroupoid $G$ of a presetoid $A$ is a maximal subpregroupoid if the dagger functor $f:G \to A$ is equivalent-on-objects, and additionally if, for every other subgroupoid $H$ of $A$ with a faithful embedding-on-objects dagger functor $g:H \to A$, the type of faithful embedding-on-objects dagger functors $h:H \to G$ with an identification $\mathrm{up}(f, g, h):h \circ f = g$ is contractible.

The core pregroupoid $Core(A)$ of a setoid $A$ is the maximal subpregroupoid of $A$. Since the core is a pregroupoid, every morphism in the core is an isomorphism, and the type of morphisms between two objects $a$ and $b$ in the core of $A$ is denoted as $a \cong_A b$.

Univalent setoids

There are two ways to show that two objects of a presetoid are identified with each other: by way of the identity type, and by way of the type of isomorphisms in the core pregroupoid of a presetoid. Univalent setoids are precisely the presetoids which the two notions of identifying objects with each other coincide.

For each object $a:Ob(A)$ and $b:Ob(A)$ of a presetoid $A$, there is a function

$\mathrm{idtoiso}(a, b):(a =_A b) \to (a \cong_A b)$

from the identity type to the type of isomorphisms in the core pregroupoid $\mathrm{Core}(A)$ of $A$. A presetoid is a univalent setoid if for each object $a:Ob(A)$ and $b:Ob(A)$ the function

$\mathrm{idtoiso}(a, b):(a =_A b) \simeq (a \cong_A b)$

is an equivalence of types.

Since every type of morphisms in the core of the presetoid is a h-set, this means that every univalent setoid is an h-groupoid.