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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:

  1. What are sufficient conditions for left and right homotopies to be equivalence relations?
  2. 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)?
  3. 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.

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  • $\begingroup$ How do you describe paths and cylinders without notions of cofibration and fibration? $\endgroup$ – Kevin Carlson Sep 29 '16 at 0:37
  • $\begingroup$ @KevinCarlson A cylinder for $x$ is a splitting of the morphism $x\sqcup x\to x$ into $x\to C(x)\to x\sqcup x$ where the second map is a weak equivalence (a good fibration is one where the second map is a fibration). A model structure guarantees the existence of good cylinders, but cylinders can exist even without it, as long as you have weak equivalences. Cf. folk.uio.no/paularne/SUPh05/DS.pdf page 18. Right homotopies are of course dual. $\endgroup$ – Daniel Robert-Nicoud Sep 29 '16 at 5:43
  • $\begingroup$ Ah, OK. In the theory of categories of fibrant objects, you have to define morphisms to be homotopic if they're right homotopic after refining the domain by a weak equivalence. Both right homotopy and homotopy define equivalence relations in this case, although this doesn't sound like the kind of thing you want. $\endgroup$ – Kevin Carlson Sep 29 '16 at 5:56

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