Category equivalence

The equivalence of categories in category theory is the relationship between categories , showing that the two categories are "essentially the same." The establishment of equivalence testifies to the deep connection of the corresponding mathematical concepts and allows one to “transfer” theorems from one structure to another.

Content

1 Definition
- 1.1 Equivalent wording
2 Examples
3 Properties
4 Literature

Definition

For two categories C and D, their equivalence is given if a functor F : C → D , a functor G : D → C , and two natural isomorphisms ε: FG → I _D and η: I _C → GF are given . Here I _C : C → C and I _D : D → D are identical functors on C and D, respectively. If F and G are contravariant functors, this determines the duality of categories .

Equivalent wording

It can be shown that the functor F : C → D defines equivalence of categories if and only if it:

quite univalent and
is dense, that is, in the isomorphism class of any element d of the category D there exists an object that has the inverse image in C under the action of F.

This is the most frequently used criterion, since it does not require explicitly constructing the “inverse” functor and two natural transformations. On the other hand, although the above property guarantees the existence of equivalence, part of the data is lost, since sometimes equivalence can be carried out in different ways. Therefore, a functor F with such properties is sometimes called weak equivalence of categories .

Another formulation uses the concept of conjugate functors : F and G define equivalence of categories if and only if they are both completely univalent and conjugate.

Examples

Between Category $C$ ${\ displaystyle C}$ from one object $c$ ${\ displaystyle c}$ and one morphism $1_{c}$ ${\ displaystyle 1_ {c}}$ and category $D$ ${\ displaystyle D}$ of two objects $d_{1}$ ${\ displaystyle d_ {1}}$ , $d_{2}$ ${\ displaystyle d_ {2}}$ and four morphisms: two identical $1_{d_{1}}$ ${\ displaystyle 1_ {d_ {1}}}$ , $1_{d_{2}}$ ${\ displaystyle 1_ {d_ {2}}}$ and two isomorphisms $\alpha \colon d_{1}\to d_{2}$ ${\ displaystyle \ alpha \ colon d_ {1} \ to d_ {2}}$ , $\beta \colon d_{2}\to d_{1}$ ${\ displaystyle \ beta \ colon d_ {2} \ to d_ {1}}$ can establish equivalence, for example, take $F$ ${\ displaystyle F}$ sending $c$ ${\ displaystyle c}$ at $d_{1}$ ${\ displaystyle d_ {1}}$ and $G$ ${\ displaystyle G}$ sending everything $D$ ${\ displaystyle D}$ at $c$ ${\ displaystyle c}$ . However, for example, the category $C$ ${\ displaystyle C}$ not equivalent to a category of two objects and two identical morphisms.
Let the category $C$ ${\ displaystyle C}$ consists of one object $c$ ${\ displaystyle c}$ and two morphisms $1_{c},f\colon c\to c$ ${\ displaystyle 1_ {c}, f \ colon c \ to c}$ where $f\circ f=1$ ${\ displaystyle f \ circ f = 1}$ . Then $f$ ${\ displaystyle f}$ defines a natural isomorphism $\mathbf {I} _{C}$ ${\ displaystyle \ mathbf {I} _ {C}}$ with itself (non-trivial, since it acts on morphisms not in an identical way).
Equivalent Category $C$ ${\ displaystyle C}$ finite-dimensional real vector spaces and category $D=\mathrm {Mat} (\mathbb {R} )$ ${\ displaystyle D = \ mathrm {Mat} (\ mathbb {R})}$ (objects are natural numbers, morphisms are matrices of the corresponding dimension): functor $F\colon C\to D$ ${\ displaystyle F \ colon C \ to D}$ associates a vector space with its dimension (which corresponds to a choice in each base space).
One of the central topics of algebraic geometry is the duality of the categories of affine schemes and commutative rings . The corresponding functor sends the ring into its spectrum - a circuit formed by simple ideals .

Properties

With equivalence of categories, all “categorical” properties are preserved: for example, the property of being the initial object , monomorphism , limit, or the property of the category of being topos .

If F : C → D is an equivalence of categories and G ₁ , G _{2 are} “inverse” to F , then G ₁ and G _{2 are} naturally isomorphic.

Literature

Equivalence of categories - an article from the Mathematical Encyclopedia
MacLane S. Chapter 4. Conjugate functors // Categories for the working mathematician / Categories. from English under the editorship of V.A. Artamonova. - M .: Fizmatlit, 2004. - S. 95-128. - 352 p. - ISBN 5-9221-0400-4 .