In a category with weak equivalences we have two notions of homotopy between morphisms, namely left homotopies (via cylinder objects), and right homotopies (path objects). Here's my questions:
- What are sufficient conditions for left and right homotopies to be equivalence relations?
- Assume the conditions two homotopy relations are equivalence relations. What are sufficient conditions for them to be equivalent (i.e. two morphisms are left homotopic if, and only if they are right homotopic)?
- Assume the conditions two homotopy relations are equivalent equivalence relations. If $x,y$ are objects, define $[x,y]$ to be the hom-set from $x$ to $y$ quotiented by the equivalence relation. What are sufficient conditions for a weak equivalence $y\stackrel{\sim}{\longrightarrow}z$ to induce an isomorphism $[x,y]\stackrel{\cong}{\longrightarrow}[x,z]$?
I know that if we have a model structure on our category with weak equivalences there are various results (cf. Quillen & co.), namely (1) is true for fibrant and cofibrant objects respectively, (2) is true for fibrant and cofibrant objects, adn so is (3). But I'd like to work with the weak equivalences only, if possible.