nLab identity-on-objects functor

Contents

Contents

Definitions

Abstract definition

Definition

Two categories with the same collection of objects with an identity-on-objects functor is a category CC enriched in the arrow category Set ISet^I of Set.

See also M-category and F-category.

By components

First, we distinguish between a category, and a category structure on a collection of objects, in the same way one does between a proset and a preorder, the equivalence relation structure on a collection of objects, in (0, 1)-category theory.

Definition

A category structure on a collection of objects ObOb is a family of sets Mor(x,y)Mor(x, y) whose elements are called morphisms, indexed by objects x:Obx:Ob and y:Oby:Ob, such that

  • for each object x:Obx:Ob, a morphism id x:Mor(x,x)\mathrm{id}_x:Mor(x, x)
  • for each object x:Obx:Ob, y:Oby:Ob, and z:Obz:Ob, a function x,y,z:Mor(y,z)×Mor(x,y)Mor(x,z)\circ_{x, y, z}:Mor(y, z) \times Mor(x, y) \to Mor(x, z)

such that

  • for each object x:Obx:Ob and y:Oby:Ob and morphism f:Mor(x,y)f:Mor(x, y), f x,y,zid x=id xf \circ_{x, y, z} \mathrm{id}_x = \mathrm{id}_x and id y x,y,zf=id y\mathrm{id}_y \circ_{x, y, z} f = \mathrm{id}_y.
  • for each object x:Obx:Ob, y:Oby:Ob, z:Obz:Ob, w:Obw:Ob and morphism f:Mor(x,y)f:Mor(x, y), g:Mor(y,z)g:Mor(y, z), h:Mor(z,w)h:Mor(z, w),
    h x,z,w(g x,y,zf)=(h y,z,wg) x,y,wfh \circ_{x, z, w} (g \circ_{x, y, z} f) = (h \circ_{y, z, w} g) \circ_{x, y, w} f

Definition

A category CC is a collection ObOb with a category structure Mor(x,y)Mor(x, y) indexed by objects x:Obx:Ob and y:Oby:Ob.

Definition

Two categories with the same collection of objects AA and BB is a collection ObOb with two category structures Mor A(x,y)Mor_A(x, y) and Mor B(x,y)Mor_B(x, y) indexed by objects x:Obx:Ob and y:Oby:Ob. The categories AA and BB are defined to be (Ob,Mor A)(Ob, Mor_A) and (Ob,Mor B)(Ob, Mor_B) respectively.

Definition

An identity-on-objects functor between two categories with the same collection of objects AA and BB is a family of functions F Mor(x,y):Mor A(x,y)Mor B(x,y)F_{Mor}(x, y):Mor_A(x, y) \to Mor_B(x, y) which preserve the category structure on the type families:

  • composition of morphisms is preserved: for all morphisms f:Mor A(x,y)f:Mor_A(x, y) and g:Mor B(y,z)g:Mor_B(y, z), F Mor(x,z)(g x,y,z Af)=F Mor(y,z)(g) x,y,z BF Mor(x,y)(f)F_{Mor}(x, z)(g \circ^A_{x, y, z} f) = F_{Mor}(y, z)(g) \circ^B_{x, y, z} F_{Mor}(x, y)(f),

  • identity morphisms are preserved: F Mor(x,x)(id x A)=id x BF_{Mor}(x, x)(id^A_x) = id^B_x.

Historical note

The notion of two categories with the same collection of objects

What does it mean for two categories to have the same collection of objects?

The current definition defines such a structure as having one collection of object and two category structures. In type theory, this could equivalently be defined as actually two categories AA and BB whose collections of objects are judgmentally equal to each other: Ob AOb BOb_A \equiv Ob_B.

Historically, authors have tried to use propositional equality to define two categories with the same collection of objects. However, this runs into a number of problems:

Hence, the notion of “two categories with the same collection of objects” is somewhat of a red herring, because it is defined as an object with two category structures, rather than two categories with the same collection of objects.

Is the identity-on-objects functor a functor?

Historically, the notion of identity-on-objects functor in between two categories with the same collection of objects was defined to be an actual functor, additionally having a function on the objects F Ob:Ob COb CF_{Ob}:Ob_C \to Ob_C which is the identity function on Ob COb_C. This was where the original name “identity-on-objects functor” comes from. The problem with this definition is twofold:

As a result, the current definition of an identity-on-objects functor, valid in all foundations and respecting the principle of equivalence, was adopted, which doesn’t have such an identity function on the objects. However, the name “identity-on-objects functor” then becomes a red herring, since it is not actually a functor between the two objects collections.

Contravariant endofunctor which is the identity-on-objects

Sometimes we want an identity-on-objects functor which behaves like a contravariant endofunctor. In this case, one does not need the notion of “two categories with the same collection of objects”.

Definition

A contravariant endofunctor which is the identity-on-objects on a category CC is a family of functions F Mor(x,y):Mor(x,y)Mor(y,x)F_{Mor}(x, y):Mor(x, y) \to Mor(y, x) such that

  • composition of morphisms is reversed: for all morphisms f:Mor A(x,y)f: Mor_A(x, y) and g:Mor B(y,z)g:Mor_B(y, z), F Mor(x,z)(g x,y,zf)=F Mor(x,y)(f) x,y,zF Mor(y,z)(g)F_{Mor}(x, z)(g \circ_{x, y, z} f) = F_{Mor}(x, y)(f) \circ_{x, y, z} F_{Mor}(y, z)(g),

  • identity morphisms are preserved: F Mor(x,x)(id x)=id xF_{Mor}(x, x)(id_x) = id_x.

This is used in particular for defining dagger categories and groupoids.

Examples

  • A dagger category is a category 𝒞\mathcal{C} with an identity-on-objects contravariant endofunctor () A,B:Mor(A,B)Mor(B,A)(-)^{\dagger_{A, B}}:Mor(A,B) \to Mor(B, A), such that for all objects A:𝒞A:\mathcal{C} and B:𝒞B:\mathcal{C} and morphisms f:Hom(A,B)f:Hom(A,B), (f A,B) B,A=f(f^{\dagger_{A, B}})^{\dagger_{B, A}} = f.

  • A groupoid is a category 𝒞\mathcal{C} with an identity-on-objects contravariant endofunctor () (1) A,B:Mor(A,B)Mor(B,A)(-)^{(-1)_{A, B}}:Mor(A,B) \to Mor(B, A) such that for all objects A:𝒞A:\mathcal{C} and B:𝒞B:\mathcal{C} and morphisms f:Hom(A,B)f:Hom(A,B), ff (1) A,B=id Bf \circ f^{(-1)_{A, B}} = id_B and f (1) A,Bf=id Af^{(-1)_{A, B}} \circ f = id_A.

Last revised on August 29, 2022 at 23:29:11. See the history of this page for a list of all contributions to it.