Homotopy Type Theory
functor (Rev #4, changes)

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Let A \xymatrix{ A &A and BB be precategories. A functor F:ABF : A \to B consists of

  • A function F 0:A 0B 0F_0 : A_0 \to B_0
  • For each a,b:Aa,b:A, a function F a,b:hom A(a,b)hom B(Fa,Fb)F_{a,b}:hom_A(a,b) \to hom_B(F a,F b), generally also denoted FF.
  • For each a:Aa:A, we have F(1 a)=1 FaF(1_a)=1_{F a}.
  • For each a,b,c:Aa,b,c: A and f:hom A(a,b)f:hom_A(a,b) amd g:hom A(b,c)g:hom_A(b,c), we have
F(gf)=FgFfF(g \circ f) = F g \circ F f


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

Composition of functors

For functors F:ABF:A\to B and G:BCG:B \to C, their composite GF:ACG \circ F : A \to C is given by

  • The composite (G 0F 0):A 0C 0(G_0 \circ F_0): A_0 \to C_0
  • For each a,b:Aa,b:A, the composite
    (G Fa,FbF a,b):hom A(a,b)hom C(GFa,GFb)(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(GF)=(HG)FH(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:

(KH)(GF) ((KH)G)F K(H(GF)) (K(HG))F K((HG)F) <!-- Created with SVG-edit - https://github.com/SVG-Edit/svgedit--> Layer 1 \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] }

See also

Category theory natural transformation full functor faithful functor


HoTT Book

category: category theory

Revision on September 4, 2018 at 18:54:58 by Ali Caglayan. See the history of this page for a list of all contributions to it.