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 $\Lambda _{i}^{n}\subset \Delta ^{n},0\leq i<n$ .^[2] A right fibration is one with the right lifting property with respect to the horn inclusions $\Lambda _{i}^{n}\subset \Delta ^{n},0<i\leq n$ .^[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". Homology, Homotopy and Applications. 12 (2): 211–230. doi:10.4310/HHA.2010.v12.n2.a7. ISSN 1532-0081.
^ a b 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)" (PDF).

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[1] Raptis, George (2010). "Homotopy theory of posets". Homology, Homotopy and Applications. 12 (2): 211–230. doi:10.4310/HHA.2010.v12.n2.a7. ISSN 1532-0081.

[Definition-2] Lurie 2009, Definition 2.0.0.3

[3] Beke, Tibor (2008). "Fibrations of simplicial sets". arXiv:0810.4960 [math.CT].