nLab fibrant 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 into it. These are the fibrant objects. Hence fibrant 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 fibrant object “has all the operations that it is supposed to”. From this perspective, a general object of a model category is a sort of generalized “presentation” of an algebraic gadget, on which some of the intended operations may be defined partially, but not necessarily all. A fibrant object is then one where all the operations are defined. This explains why fibrant objects are good for mapping into: if the values of some operations are missing in an object $X$, then some morphisms into $X$ that “ought to exist” may fail because they ought to take image in the missing values of operations.

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

The adjective “fibrant” is also used in contexts more general than these, but with a similar intuition. For instance, there are fibrant types in two-level type theory, and sometimes double categories with all companions and conjoints are called “fibrant” (or “bifibrant”).

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

Definition

In a model category

In a model category, an object $X$ is said to be fibrant if the unique morphism $X\to 1$ to the terminal object is a fibration.

$$(X \;\; \text{fibrant}) \;\;\Leftrightarrow\;\; ( X \overset{\in Fib}{\longrightarrow} 1)$$

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

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

$$X \overset{\in Cof \cap W}{\longrightarrow} X_{fib} \overset{\in Fib}{\longrightarrow} 1$$

Examples

1. In the classical model structure on topological spaces, every object $X$ is fibrant (namely the continuous function $X \to \ast$ to the point space is a Serre fibration).

2. In the classical model structure on simplicial sets, the fibrant objects are the Kan complexes.

3. In the canonical model structure on Cat, every object is fibrant, and similarly in other categorical examples.

Related concepts

• cofibrant object, bifibrant object

• fibration, cofibration

• fibrant replacement

• fibrant type

• functorial factorization