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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 21st 2012
   • (edited Feb 21st 2012)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI was looking for the simplest (explicit) proof that a quasi-category is a (infinity,1)-category, i.e. that 2-cells and above are equivalences. All of our entries more or less assume this rather than indicating a proof, so then I looked but could not find an explicit reference, in the nLab pages, to a proof in the literature. What is the best source for that?

   I was looking for the simplest (explicit) proof that a quasi-category is a (infinity,1)-category, i.e. that 2-cells and above are equivalences. All of our entries more or less assume this rather than indicating a proof, so then I looked but could not find an explicit reference, in the nLab pages, to a proof in the literature. What is the best source for that?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 21st 2012
   • (edited Feb 21st 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt's the same proof as for Kan complexes: exhibiting the inverse of a 2-cell only needs filling inner horns.

   It’s the same proof as for Kan complexes: exhibiting the inverse of a 2-cell only needs filling inner horns.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 21st 2012
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI knew it for 2-cells (and so can manufacture a proof in higher dimensions). I was partially pointing out that even this seems not to be given in a prominent position in 'the literature'. I looked for it in both Joyal's notes and in Lurie, (not a thorough search in either case) and could not find it. It also relates to the query that is still there at the foot of [[quasi-category]].

   I knew it for 2-cells (and so can manufacture a proof in higher dimensions). I was partially pointing out that even this seems not to be given in a prominent position in ’the literature’. I looked for it in both Joyal’s notes and in Lurie, (not a thorough search in either case) and could not find it. It also relates to the query that is still there at the foot of quasi-category.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 21st 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi Tim, in Lurie's book this is discussed in section 1.2.2 "Mapping spaces in higher category theory". Proposition 1.2.2.3 there is the statement that you are after.

   Hi Tim,

   in Lurie’s book this is discussed in section 1.2.2 “Mapping spaces in higher category theory”. Proposition 1.2.2.3 there is the statement that you are after.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 21st 2012
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThanks that is a great help.

   Thanks that is a great help.