In mathematics, specifically in homotopy theory in the context of a model category M, a fibrant objectA of M is an object that has a fibration to the terminal object of the category.
model category M, a fibrantobject A of M is an object that has a fibration to the terminal object of the category. The fibrantobjects of a closed model...
and are therefore of fundamental importance. Kan complexes are the fibrantobjects in this model category. The name is in honor of Daniel Kan. For each...
isomorphism if there exists an inverse of f. Kan complex A Kan complex is a fibrantobject in the category of simplicial sets. Kan extension 1. Given a category...
theory. In any model category, a weak equivalence between cofibrant-fibrantobjects is a homotopy equivalence. J. H. C. Whitehead, Combinatorial homotopy...
the subcategory of “good” (fibrant or cofibrant) objects.* By first taking a fibrant or cofibrant resolution of an object and then applying that functor...
{\displaystyle C\cap W.} An object X {\displaystyle X} is called fibrant if the morphism X → 1 {\displaystyle X\rightarrow 1} to the terminal object is a fibration...
the category of simplicial sets are known as Kan fibrations, and the fibrantobjects are known as Kan complexes. Some of Kan's later work concerned model...
is fibrant and there is a weak equivalence from X to Z then Z is said to be a fibrant replacement for X. In general, not all objects are fibrant or cofibrant...
(right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived functor theorem of Quillen says that the total left...
influential paper that defines Browns categories of fibrantobjects and dually Brown categories of cofibrant objects 1974 Shiing-Shen Chern–James Simons Chern–Simons...
_{*}F\to \pi _{*}G} is an isomorphism. A sheaf of spectra is then a fibrant/cofibrant object in that category. The notion is used to define, for example, a...
been developed which partition their types into fibrant types, which respect paths, and non-fibrant types, which do not. Cartesian cubical computational...
symmetric spectra have quite different behaviour: in S-modules every object is fibrant (which is not true in symmetric spectra), while in symmetric spectra...
equivalence (of simplicial sets) for any C-local object W. An object W is called C-local if it is fibrant (in M) and s ∗ : map ( B , W ) → map ( A ,...
keeping the set of states constant. All objects of the model categories of flows and multipointed d-spaces are fibrant. It can be checked that the cylinders...
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