Are all isomorphisms contained in the class of weak equivalences?

Question

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...

Answer 1

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 $x$.