Homotopy Type Theory functor > history (Rev #4)

Definition

Let $A$ and $B$ be precategories. A functor $F : A \to B$ 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

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

HoTT Book