Duality (category theory)

Duality in category theory is the relationship between the properties of category C and the so-called dual properties of the dual category C ^op . Taking the statement concerning category C and swapping the image and prototype of each morphism, as well as the order of application of morphisms, we obtain a dual statement regarding category C ^op . The principle of duality consists in the fact that true statements after such an operation turn into true, and false statements into false ones.

Formal definition

The language of category theory is defined as a first-order language with two types of symbols - objects and morphisms, with the property of an object to be an image or a prototype of morphism, and also with a symbol for the composition of morphisms.

Let σ be any word of a language. The dual word σ ^{op is} formed by the following rules:

swap all “images” for “types” in σ ,
reverse the order of the composition of morphisms, that is, all occurrences $g\circ f$ ${\ displaystyle g \ circ f}$ replaced by $f\circ g$ ${\ displaystyle f \ circ g}$ .

In other words, it is necessary to turn all arrows and rearrange the arguments of all compositions .

Duality is the observation that σ holds in some category C if and only if σ ^{op is} made in C ^op .

Examples

Morphism $f\colon A\to B$ ${\ displaystyle f \ colon A \ to B}$ - monomorphism , when of $f\circ g=f\circ h$ ${\ displaystyle f \ circ g = f \ circ h}$ follows $g=h$ ${\ displaystyle g = h}$ . Applying the operation of duality, we obtain the statement that $g\circ f=h\circ f$ ${\ displaystyle g \ circ f = h \ circ f}$ follows $g=h$ ${\ displaystyle g = h}$ . For morphism $f\colon B\to A$ ${\ displaystyle f \ colon B \ to A}$ , it means exactly that f is an epimorphism . Thus, the property “to be a monomorphism” is dual to the property “to be an epimorphism”.
The limit and the limit are dual concepts.
The initial object and the terminal object are dual concepts.

Literature

The dual category is an article from the Mathematical Encyclopedia . Danilov M.E.
Duality principle - an article from the Mathematical Encyclopedia . Kabakov F.A., Parkhomenko A.S., Wojciechowski M.I., Fofanova T.S.
Duality is an article from the Mathematical Encyclopedia . Dolgachev I.V. and others.
MacLane S. Chapter 2. Constructions in categories // Categories for the working mathematician = Categories for the working mathematician / Trans. from English by ed. V.A. Artamonov. - M .: Fizmatlit, 2004. - P. 43-67. - 352 s. - ISBN 5-9221-0400-4 .

Duality (category theory)

Formal definition

Examples

Literature

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