mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically in

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...

in the context of a

model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstrac ...

''M'', a fibrant object ''A'' of ''M'' is an

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

that has a

fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...

to the

terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...

of the

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

Properties

The fibrant objects of a

closed model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract ...

are characterized by having a

right lifting property In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given c ...

with respect to any trivial cofibration in the category. This property makes fibrant objects the "correct" objects on which to define

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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...

s. In the context of the theory of

simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...

s, the fibrant objects are known as Kan complexes after

Daniel Kan Daniel Marinus Kan (or simply Dan Kan) (August 4, 1927 – August 4, 2013) was a Dutch mathematician working in category theory and homotopy theory. He was a prolific contributor to both fields for six decades, having authored or coauthored sever ...

. They are the

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 structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...

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.

References

*P.G. Goerss and J.F. Jardine, ''Simplicial Homotopy Theory'', Progress in Math., Vol. 174, Birkhauser, Boston-Basel-Berlin, 1999. . Homotopy theory Objects (category theory) {{geometry-stub