# Homotopy Type Theory functor (Rev #4, changes)

Showing changes from revision #3 to #4: Added | Removed | Changed

## Definition

Let  \xymatrix{ A &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
$F(g \circ f) = F g \circ F f$

## 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:

 \array{ && (K H)(G F) \\ & \nearrow && \searrow \\ ((K H) G) F && && K (H (G F)) \\ \downarrow && && \uparrow \\ (K(H G)) F && \longrightarrow && K( (H G) F) } \begin{svg} <svg width="640" height="480" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg"> } <!-- Created with SVG-edit - https://github.com/SVG-Edit/svgedit--> <g class="layer"> <title>Layer 1</title> <path d="m181.62251,199.62251l132.82468,-96.63715l132.82499,96.63715l-50.7344,156.36285l-164.18071,0l-50.73455,-156.36285z" fill="none" id="svg_3" stroke="#000000" stroke-width="5"/> </g> </svg> \end{svg}
\xymatrix{ & K(H(GF)) \ar@{=}[dl] \ar@{=}[dr]\\ (KH)(GF) \ar@{=}[d] && K((HG)F) \ar@{=}[d]\\ ((KH)G)F && (K(HG))F \ar@{=}[ll] }