nLab cofibrant object

Idea

A model category is a homotopical category equipped with especially nice control over the weak equivalences. In particular every object of the category is weakly equivalent to an object that is particularly well-behaved for forming derived hom-spaces out of it. These are the cofibrant objects. Hence cofibrant objects are particularly good representatives of objects, which are the “same” as the given objects up to weak equivalence.

One general intuition is that a cofibrant object is one that’s uncoiled, unwound or puffed up. An object oftens fails to be cofibrant when it’s “coiled up too tightly”, with a bunch of strict equations that should be equivalences or paths. This explains why cofibrant objects are good for mapping out of: if some equations in $X$ hold too strictly, then those equations would have to be preserved by a (strict) map out of $X$, which might not be possible if the intended codomain doesn’t have such strict equations.

This concept also exists in homotopical categories with less extra structure than that of a full model category. For instance a cofibration category implements roughly half of the model category axioms, namely those for cofibrations, and it has a concept of weakly equivalent replacement by a cofibrant object.

The adjective “cofibrant” is also used in contexts more general than these, but with a similar intuition. For instance, there are cofibrant types? in two-level type theory.

The dual concept is a fibrant object: an object which is good for mapping into. These also always exist in model categories, but not necessarily in cofibration categories (though they do in the dual notion of fibration categories).

Definition

In a model category

In a model category, an object $X$ is said to be cofibrant if the unique morphism $0\to X$ from the initial object is a cofibration.

$$(X \;\; \text{cofibrant}) \;\;\Leftrightarrow\;\; ( 0 \overset{\in Cof}{\longrightarrow} X)$$

Hence the axiom that every morphism in a model category factors as a cofibration followed by an acyclic fibration implies the existence of cofibrant resolution.

Namely, for $X$ any object, the factorization of the initial morphism as a cofibration followed by an acyclic fibration yields a cofibrant object $X_{cof}$ weakly equivalent to $X$

$$0 \overset{\in Cof}{\longrightarrow} X_{cof} \overset{\in Fib \cap W}{\longrightarrow} X$$

Examples

1. In the classical model structure on topological spaces, the cofibrant objects are the retracts of cell complexes, in particular the CW-complexes.

2. In the classical model structure on simplicial sets, every object is cofibrant.

3. In the canonical model structure on Cat, every object is cofibrant. However, this ceases to be the case in other canonical categorical model structures. For instance, in the canonical model structure on 2-categories, the cofibrant objects are those whose underlying 1-category is freely generated by a quiver.

4. In the projective model structure on dgc-algebras in non-negative degree, the cofibrant objects are the Sullivan algebras (see there). This plays a key role in rational homotopy theory.

Related concepts

• fibrant object, bifibrant object

• fibration, cofibration

• cofibrant replacement

• cofibrant type?

• functorial factorization