# Fibrant object

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Fibrant_object
> Markdown URL: https://mediated.wiki/source/Fibrant_object.md
> Source: https://en.wikipedia.org/wiki/Fibrant_object
> Source revision: 1278979692
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Mathematical concept}}
In [mathematics](/source/mathematics), specifically in [homotopy theory](/source/homotopy_theory) in the context of a [model category](/source/model_category) ''M'', a '''fibrant object''' ''A'' of ''M'' is an [object](/source/object_(category_theory)) that has a [fibration](/source/fibration) to the [terminal object](/source/terminal_object) of the [category](/source/category_(mathematics)).

==Properties==

The fibrant objects of a [closed model category](/source/closed_model_category) are characterized by having a [right lifting property](/source/right_lifting_property) with respect to any [trivial cofibration](/source/cofibration) in the category.  This property makes fibrant objects the "correct" objects on which to define [homotopy group](/source/homotopy_group)s.  In the context of the theory of [simplicial set](/source/simplicial_set)s, the fibrant objects are known as '''Kan complexes''' after [Daniel Kan](/source/Daniel_Kan). They are the [Kan fibration](/source/Kan_fibration)s over a point.

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

==References==
*P.G. Goerss and J.F. Jardine, ''Simplicial Homotopy Theory'', Progress in Math., Vol. 174, Birkhauser, Boston-Basel-Berlin, 1999.  {{ISBN|3-7643-6064-X}}.

== Further reading ==
* https://ncatlab.org/nlab/show/fibrant+object

Category:Homotopy theory
Category:Objects (category theory)

{{differential-geometry-stub}}

---
Adapted from the Wikipedia article [Fibrant object](https://en.wikipedia.org/wiki/Fibrant_object) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Fibrant_object?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.