Consider $(M,Fib,Cof,WE)$ a model category, that is a complete/cocomplete category with 3 wide subcategories satisfying certain properties.
Now consider $X\in\operatorname{Ob}M$ and the slice category $M_{/X}$ (objects are triples $(m,*,f_m:m\to X)$). Consider $U_X:M_{/X}\to M$ the forgetful functor. I have shown that $(M_{/X},U_X^{-1}(Fib),U_X^{-1}(Cof),U_X^{-1}(WE))$ is a model category.
I have two questions : Is there always a (unique) terminal/intial object in a model category ?
Can we describe fibrant and cofibrant elements more than saying their images by the forgetful functor are fibrant or cofibrant ?
Best Answer
As pointed out by Zhen Lin in the comments we have an asymmetry here. Fibrant objects in the slice category are the objects with a structure morphism which is a fibration. This is not the case for cofibrant objects where we require the structure object to be cofibrant itself. In the coslice category however we would have that the fibrant objects are the ones who have a fibrant structure object, and cofibrant objects are the ones who have a cofibration as structure morphism.