The Yoneda Lemma
subject to two coherence conditions, called associativity and identity . The associativity condition states that ( f ◦ g ) ◦ h = f ◦ ( g ◦ h ) , while the identity condition states that for every object X ∈ Ob( C ) there exists a morphism id X : X → X such that f ◦ id X = f and id X ◦ g = g for any object A and morphism f : A → X or g : X → A.
It is helpful to understand morphisms as paths between objects. With this intuition, the composition of two morphisms represents following one path to an object, and then another path from it. The requirement of associativity then states that there should only be one way that three paths A → B → C → D can be combined into a single path A → D : whether one simplifies it to A → C → D or A → B → D first, the resulting arrow A → D is the same. The identity constraint guarantees a ‘zero path’ that represents remaining still at an object. Also note that associativity allows us to write f ◦ g ◦ h unambiguously without brackets.
The structure of a category is very general, able to encompass many different mathematical notions. Arguably the most important is the category Set of sets, where the objects are sets and morphisms represent functions between sets.
Definition 2 We specify Set as follows. Ob( Set ) is the collection of all sets. For any two sets A and B, we have Hom( A,B ) = B A , the set of all functions from A to B. Composition is given by standard function composition i.e. ( f ◦ g ) x = g ( f ( x )) . Identities are simply the identity functions given by id A ( x ) = x.
Identity and associativity in this case are standard results from set theory, but the curious reader can easily confirm that the stated definition of function composition satisfies both conditions.
1.4 Key concepts 1.4.1 Isomorphisms Isomorphisms tell us that two objects in a category are essentially the same. In Set they are simply bijections, i.e. 1–1 mappings.
Definition 3 If f : X → Y is a morphism, we say that f is an isomorphism iff there exists a morphism f − 1 : Y → X such that f ◦ f − 1 = id X and f − 1 ◦ f = id Y .
If there exists an isomorphism between X and Y we say that X and Y are isomorphic .
1.4.2 Functors and the category of categories Functors allow us to transform categories in much the same way that functions map between sets. The only restriction is that they must respect the structure of the categories.
Definition 4 To specify a functor F : C → D from a category C to a category D, one specifies, for each object X ∈ Ob( C ) , an object F ( X ) ∈ Ob( D ) , and, for each morphism f : X → Y , a morphism F ( f ) : F ( X ) → F ( Y ) . To be a functor, F is subject to the coherence conditions that it preserves identity i.e. F (id X ) = id F ( X ) and that it preserves composition i.e. F ( f ◦ g ) = F ( f ) ◦ F ( g ) .
Functors have a lot in common with functions, and we can formalize this. Just as functions transform sets, functors transform categories. As with functions, we can define a composition F ◦ G by the equation
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