homotopy invariant functor and homotopic maps

1. Proposition

Let 20240218-homotopy_invariant_functor_and_factorization_of_htp_760824120891fc757bc0445145b027d09b237baa.svg be a category, a functor.

TFAE:

is homotopy equivalent as in factorization through Category hTop
for homotopic maps it follows, that

20240218-homotopy_invariant_functor_and_factorization_of_htp_a584ec33841f7b82d20483e93b8de3043cc6ee19.svg

2. Proof

2.1. 1) 2)

Let

20240218-homotopy_invariant_functor_and_factorization_of_htp_931eabec1c60eb2ed6c47f4dfc4e496ba14725be.svg

Let 20240218-homotopy_invariant_functor_and_factorization_of_htp_7c29ddc16641e108c236f31c214bc6207f24811e.svg be homotopic maps, then

20240218-homotopy_invariant_functor_and_factorization_of_htp_7902f22924689d6a8c7d55f9f5ff7c06d7693b5d.svg

2.2. 2) 1)

Let

20240218-homotopy_invariant_functor_and_factorization_of_htp_9dbc2ff54b962532a82812e546c46485ec3043c1.svg

Then since 20240218-homotopy_invariant_functor_and_factorization_of_htp_23eb926065f84471a319a36b6cdec9bebb87542f.svg is object-injective and homotopic maps get mapped to the same morphism, this functor is welldefined Functorality follows from

20240218-homotopy_invariant_functor_and_factorization_of_htp_19f5993d74448993d6b53d2256a34c10717338fc.svg

and

20240218-homotopy_invariant_functor_and_factorization_of_htp_7cc3e1256ffb106bf8ee34806bd031657e848846.svg