Subject: Re: cofibrant objects
Date: 22 Apr 2003 10:58:49 -0400
From: "Mark W. Johnson"
To: "(Don Davis)"
I have two different tricks for finding non-cofibrant final objects.
1) A new kind of cheat:
Take your favorite bicomplete category and choose both fibrations and
weak equivalences as any morphisms. I know the intersection of the
three distinguished classes in a model structure must be the
isomorphisms, so the cofibrations are forced to be the isomorphisms.
Unless I'm missing something, this is a model structure and the only
cofibrant object is the initial object (only defined up to isomorphism
anyway). Hence, any bicomplete category with different initial and
final objects should yield an example.
2) A variant on Goodwillie's response about objects between:
Pick your favorite model category and your favorite object in it. I'll
call the object C. Then the category of objects over C, sometimes
called a slice category, inherits a model structure from the original
category. Here all three classes of maps are defined as in the original
category and the slight surprise is that factorizations and lifts in the
original category automatically lie in this over-category.
Now, it is easy to see that C has become the final object and that the
notion of cofibrant remains unaltered. In other words, choosing C which
is not cofibrant in the original category will give an example where the
final object is not cofibrant.
For those who consider this construction artificial, notice that any
"more natural" example will also be of this form; choosing C to be the
already existing final object just recovers the original category. The
same is true for Tom's suggestion if we take f to be the unique map from
the initial object to the final object, which will fail to be a
cofibration precisely when the final object is not cofibrant. (Also, my
construction is always the special case where Tom's morphism starts with
the initial object.)
For a specific example, pick your favorite non-Kan simplicial complex
and the category of arrows emanating from it. The opposite of this
category is the type of thing I'm talking about.