0
$\begingroup$

Suppose $\mathscr{C}$ is a model category, defined as in the book Simplicial homotopy theory by Goerss and Jardine, chapter $2$.

Is it true that every isomorphism in $\mathscr{C}$ is a weak equivalence?

If we assume that for every object $x\in\mathscr{C}$ there is a weak equivalence that has $x$ either as source or target, then it is easy to prove that this is true. Indeed, if e.g. $f:x\to y$ is a w.e., then $f=f1_x$, so that by $2$-out-of-$3$ the identity $1_X$ is a w.e., and then we can write every isomorphism as a retract of the identity. However, I don't know how to prove this fact without making any further assumptions.

Also, in all the examples I have in mind it is the case that weak equivalences contain isomorphisms. However, not containing them would not be a huge problem, I think, as when we localize isomorphisms will remain isomorphisms...

$\endgroup$
1
$\begingroup$

If we assume that for every object $x\in\mathscr{C}$ there is a weak equivalence that has $x$ either as source or target, then it is easy to prove that this is true.

By axiom $\mathbf{CM5}$ on page 72, we have in particular that $\mathrm{id}_x : x \to x$ factors as a fibration followed by a trivial cofibration; the latter is a weak equivalence whose codomain is $x$.

$\endgroup$
  • $\begingroup$ Ah, great! Thanks, I didn't consider applying CM5. $\endgroup$ – Daniel Robert-Nicoud Apr 3 '18 at 23:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.