Joyal's theorem

In mathematics, Joyal's theorem is a theorem in homotopy theory that provides necessary and sufficient conditions for the solvability of a certain lifting problem involving simplicial sets. In particular, in higher category theory, it proves the statement "an ∞-groupoid is a Kan complex", which is a version of the homotopy hypothesis.{{harvnb|Cisinski|2023|loc=Theorem 3.5.1.}}

The theorem was introduced by André Joyal.

Joyal extension theorem

Let $C$ be quasicategory and let $u:X \rightarrow Y$ be a morphism of $C$ . The following conditions are equivalent:{{harvnb|Theorem 4.4.2.6 in Kerodon}}{{harvnb|Rezk|2022||loc=34.2. Theorem}}{{harvnb|Lurie|2009||loc=Proposition 1.2.4.3}}{{harvnb|Joyal|2002|loc=Theorem 1.3}}

(1)The morphism $u$ is an isomorphism.

(2)Let $n \geq 2$ and let $\sigma_0 : \Lambda^n_0 \to C$ be a morphism of simplicial sets for which the initial edge

$\Delta^1 \cong N_{\bullet }( \{ 0 < 1\} ) \rightarrow \Lambda^n_ n \xrightarrow {\sigma _0} C$

is equal to $u$ . Then $\sigma_0$ can be extended to an n-simplex $\sigma : \Delta^n \to C$ .

(3)

Let $n \geq 2$ and let $\sigma_0 : \Lambda^n_n \rightarrow C$ be a morphism of simplicial sets for which the initial edge

$\Delta^1 \cong N_{\bullet }( \{ n-1 < n\} ) \rightarrow \Lambda^n_ n \xrightarrow {\sigma _0} C$

is equal to $u$ . Then $\sigma_0$ can be extended to an n-simplex $\sigma : \Delta^n \to C$ .

Joyal lifting theorem

Let $p: C \rightarrow D$ be an inner fibration (Joyal used mid-fibration{{harvnb|Lurie|2009||page=xiv}}) between quasicategories, and let $f \in C_1$ be an edge such that $p (f)$ is an isomorphism in $D$ . The following are equivalent:{{harvnb|Rezk|2022|loc=34.17. Theorem (Joyal lifting).}}{{harvnb|Haugseng|loc=Theorem 5.3.1.}}{{harvnb|Kapulkin|Voevodsky|2020|loc=Theorem 2.10}}{{harvnb|Land|2021|loc=Theorem. 2.1.8}}{{harvnb|Joyal|2002|loc=Theorem 2.2}}{{harvnb|Joyal|2008|loc=Theorem 6.13}}

(1) The edge $f$ is an isomorphism in $C$ .

(2) For all $n \geq 2$ , every diagram of the form

File:Joyal lifting theorem.svg

admits a lift.

(3)For all $n \geq 2$ , every diagram of the form

File:Joyal lifting theorem 2.svg

admits a lift.

Notes

References

{{cite journal |doi=10.1016/S0022-4049(02)00135-4 |title=Quasi-categories and Kan complexes |date=2002 |last1=Joyal |first1=A. |journal=Journal of Pure and Applied Algebra |volume=175 |issue=1–3 |pages=207–222 }}
{{cite web |last1=Rezk |first1=Charles |title=Introduction to quasicategories |url=https://ncatlab.org/nlab/files/Rezk-IntroToQuasicategories.pdf|date=2022|via=ncatlab.org}}
{{cite book |doi=10.1007/978-3-030-61524-6_2 |chapter=Joyal's Theorem, Applications, and Dwyer–Kan Localizations|chapter-url={{Google books|1sMqEAAAQBAJ|page=99|plainurl=yes}} |title=Introduction to Infinity-Categories |series=Compact Textbooks in Mathematics |date=2021 |last1=Land |first1=Markus |pages=97–161 |isbn=978-3-030-61523-9|zbl= 1471.18001 }}
{{cite journal |doi=10.1112/topo.12173 |title=A cubical approach to straightening |date=2020 |last1=Kapulkin |first1=Krzysztof |last2=Voevodsky |first2=Vladimir |journal=Journal of Topology |volume=13 |issue=4 |pages=1682–1700 }}
{{Cite web |title=Theorem 4.4.2.6 (Joyal) |url=https://kerodon.net/tag/019F |website=Kerodon|ref={{harvid|Theorem 4.4.2.6 in Kerodon}}}}
{{Cite web |title=Proposition 4.4.2.13. |url=https://kerodon.net/tag/01H0 |website=Kerodon|ref={{harvid|Proposition 4.4.2.13. in Kerodon}}}}
{{cite web |last1=Haugseng |first1=Rune |title=Introduction to ∞-Categories |url=https://people.math.rochester.edu/faculty/doug/otherpapers/Haugseng-intro.pdf}}
{{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2023|publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}
{{cite book |last=Lurie |first=Jacob|author-link=Jacob Lurie|date=2009|title=Higher Topos Theory|arxiv=math/0608040|publisher=Princeton University Press|isbn=978-0-691-14048-3}}
{{cite web |first1=André |last1=Joyal |title=THE THEORY OF QUASI-CATEGORIES (Vol I) Draft version |url=https://math.uchicago.edu/~may/PEOPLE/JOYAL/0newqcategories.pdf |date=2008}}

Joyal's theorem

Joyal extension theorem

Joyal lifting theorem

Notes

References

Further reading