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2 cut out boring part of question

Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent references have strengthened this axiom by assuming that this is actually a functorial factorization.

If you take a "classical" model category (not exactly in Quillen's sense - I would still like to assume that it has all small limits and colimits), it becomes natural to ask whether it has such a functorial factorization. More, one might wonder how canonical this factorization is, even just on the level of objects.

So in this situation, let's say that we have either:

• a very large category whose objects are "factorization functors", and whose morphisms are natural transformations between them (of necessity natural weak equivalences), or
• a very large category whose objects are functorial cofibrant replacements for objects, and whose morphisms are again natural weak equivalences.

Are there easy examples where these categories are empty? Are there examples where they are nonempty, but the functorial factorization is "noncanonical" in the sense that the category of factorizations is noncontractible? Does this category of factorizations become contractible under stronger assumptions (such as cofibrant generation)?

Added later: It turns out that the question of homotopy type has a boring answer (and, embarassingly, one answerable by the standard techniques). Let's suppose we have one factorization functor $F$, so that any arrow $g:x \to y$ factors canonically as $$x \stackrel{c(g)}{\to} F(g) \stackrel{f(g)}{\to} y.$$ Then, naturally associated to any other such functor $G$, we get a third replacement functor abusively written as $G \circ F$, obtained by applying $G$ to the map $c(g)$; this gives a factorization $$x \to (G\circ F)(g) \to F(g) \to y$$ and the first map, together with the composite of the latter two maps, gives a new factorization.

There are natural transformations of factorization functors $F \leftarrow G \circ F \rightarrow G$, and this is natural in $G$; this provides a two-step homotopy contracting the space of factorization functors down to the constant $F$.

(However, the responses to the first question have already informed me quite a bit!)

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# How canonical is cofibrant replacement?

Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent references have strengthened this axiom by assuming that this is actually a functorial factorization.

If you take a "classical" model category (not exactly in Quillen's sense - I would still like to assume that it has all small limits and colimits), it becomes natural to ask whether it has such a functorial factorization. More, one might wonder how canonical this factorization is, even just on the level of objects.

So in this situation, let's say that we have either:

• a very large category whose objects are "factorization functors", and whose morphisms are natural transformations between them (of necessity natural weak equivalences), or
• a very large category whose objects are functorial cofibrant replacements for objects, and whose morphisms are again natural weak equivalences.

Are there easy examples where these categories are empty? Are there examples where they are nonempty, but the functorial factorization is "noncanonical" in the sense that the category of factorizations is noncontractible? Does this category of factorizations become contractible under stronger assumptions (such as cofibrant generation)?