Left homotopy in the quillen model structure

Question

Let $ \mathscr{T} $ be the category of [insert technical conditions here] topological spaces. Equip $ \mathscr{T}$ with the quillen model structure. The category of based spaces $ \mathscr{T}_*$ inherits a model structure.

• Question 1: Let $ f,g : X \to Y $ be continuous maps in $ \mathscr{T}$. Is it true that $f$ and $g$ are left homotopic in a model category sense iff $f$ and $g$ are homotopic in an elementary sense?
• Question 2: Let $f,g : X \to Y $ be based maps in $ \mathscr{T}_*$ between cofibrant objects. Is it true that $f $ and $g$ are left homotopic in a model category sense iff $ f $ and $g $ are pointed homotopic in an elementary sense?

Answer 1

This is dealt with in the book ``More concise algebraic topology'' by Kate Ponto and myself, and of course elsewhere, I am sure. We used compactly generated spaces (for us, that means weak Hausdorff $k$-spaces). Since all spaces and all based spaces in the Quillen model structure are fibrant, 14.3.9 shows that the three kinds of left homotopies one can define in any model category all give the same notion of homotopic maps. The choices depend on how good you require your cylinder objects to be. In the case of based spaces your elementary sense presumably means via homotopies defined on the cylinders $X\wedge I_+$. Here there is the quibble that the natural map $X\wedge I_+\longrightarrow X$ must be a weak equivalence for $X\wedge I_+$ to be a model theoretic cylinder object. This holds if you take your based spaces to be well-pointed (alias nondegenerately based, alias cofibrant in the Hurewicz model structure). Then 14.3.9 kicks in as before; you do not need to restrict to cofibrant based spaces. The question of using a standard elementary notion of homotopy is axiomatized in 16.4.10 and 16.4.11 of our book (and to the best of my knowledge nowhere else).