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Idea

The concept if functor is the evident concept of homomorphisms between categories.

Definition

Let $A$ and $B$ be precategories . A Informally, afunctor $F : A \to B$ consists of

• A function $F_0 : A_0 \to B_0$
• For each $a,b:A$, a function $F_{a,b}:hom_A(a,b) \to hom_B(F a,F b)$, generally also denoted $F$.
• For each $a:A$, we have $F(1_a)=1_{F a}$.
• For each $a,b,c: A$ and $f:hom_A(a,b)$ amd $g:hom_A(b,c)$, we have

  Definition functor_data (C C' : precategory_ob_mor) := total2 (
      fun F : ob C -> ob C' => 
               forall a b : ob C, a --> b -> F a --> F b).

  Definition functor_on_objects {C C' : precategory_ob_mor}
       (F : functor_data C C') :  ob C -> ob C' := pr1 F.
  Coercion functor_on_objects : functor_data >-> Funclass.

  Definition functor_on_morphisms {C C' : precategory_ob_mor} (F : functor_data C C') 
      { a b : ob C} : a --> b -> F a --> F b := pr2 F a b.

  Local Notation "# F" := (functor_on_morphisms F)(at level 3).


  Definition is_functor {C C' : precategory_data} (F : functor_data C C') :=
       dirprod (forall a : ob C, #F (identity a) == identity (F a))
               (forall a b c : ob C, forall f : a --> b, forall g : b --> c, 
                  #F (f ;; g) == #F f ;; #F g).

  Lemma isaprop_is_functor (C C' : precategory_data) 
        (F : functor_data C C'): isaprop (is_functor F).
  Proof.
    apply isofhleveldirprod.
    apply impred; intro a.
    apply (pr2 (_ --> _)).
    repeat (apply impred; intro).
    apply (pr2 (_ --> _)).
  Qed.

  Definition functor (C C' : precategory) := total2 (
    fun F : functor_data C C' => is_functor F).

Properties

By induction on identity, a functor also preserves $idtoiso$ (See precategory).

Composition of functors

For functors $F:A\to B$ and $G:B \to C$, their composite $G \circ F : A \to C$ is given by

• The composite $(G_0 \circ F_0): A_0 \to C_0$
• For each $a,b:A$, the composite
  $$(G_{F a, F b} \circ F_{a,b}):hom_A(a,b)\to hom_C(G F a, G F b)$$

Lemma 9.2.9

Composition of functors is associative $H(G F)=(H G)F$.

Proof: Since composition of functions is associative, this follows immediately for the actions on objects and on homs. And since hom-sets are sets, the rest of the data is automatic. $\square$

Lemma 9.2.10

Lemma 9.2.9 is coherent, i.e. the following pentagon of equalities commutes:

See also

Category theory natural transformation full functor faithful functor

References