In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in homotopy theory in the context of a model category

Model category

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...

 M, a fibrant object A of M is an object that has a fibration

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...

 to the terminal object of the category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

.

Properties

The fibrant objects of a closed model category are characterized by having a right lifting property

Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space E to another one, B...

 with respect to any trivial cofibration

Cofibration

In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...

 in the category. This property makes fibrant objects the "correct" objects on which to define homotopy group

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...

s. In the context of the theory of simplicial set

Simplicial set

In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...

s, the fibrant objects are known as Kan complexes after Daniel Kan

Daniel Kan

Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an...

. They are the Kan fibration

Kan fibration

In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category for simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category...

s over a point.

Dually is the notion of cofibrant object, defined to be an object $c$ such that the unique morphism $\varnothing\to c$ from the initial object to $c$ is a cofibration.