Fibration of simplicial sets
In mathematics, especially in homotopy theory,[1] a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions [math]\displaystyle{ \Lambda^n_i \subset \Delta^n, 0 \le i \lt n }[/math].[2] A right fibration is one with the right lifting property with respect to the horn inclusions [math]\displaystyle{ \Lambda^n_i \subset \Delta^n, 0 \lt i \le n }[/math].[2] A Kan fibration is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is both a left and right fibration.[3] On the other hand, a left fibration is a coCartesian fibration and a right fibration a Cartesian fibration. In particular, category fibered in groupoids over another category is a special case of a right fibration of simplicial sets in the ∞-category setup.
References
- ↑ Raptis, George (2010). "Homotopy theory of posets" (in EN). Homology, Homotopy and Applications 12 (2): 211–230. doi:10.4310/HHA.2010.v12.n2.a7. ISSN 1532-0081. https://www.intlpress.com/site/pub/pages/journals/items/hha/content/vols/0012/0002/a007/abstract.php.
- ↑ 2.0 2.1 Lurie 2009, Definition 2.0.0.3
- ↑ Beke, Tibor (2008). "Fibrations of simplicial sets". arXiv:0810.4960 [math.CT].
- Ch. 2 of Lurie's Higher Topos Theory.
- Lurie, J. (2009). "Lecture 9 of Algebraic K-Theory and Manifold Topology (Math 281)". https://www.math.ias.edu/~lurie/281notes/Lecture9-Fibrations2.pdf.
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