Geodesic
A relative 2–nerve
Abellán García, Fernando1 ; Dyckerhoff, Tobias1 ; Stern, Walker H2
1Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
2Department of Mathematics, University of Virginia, Charlottesville, VA, United States
Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 3147-3182
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We introduce a 2–categorical variant of Lurie’s relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to ∞–categorical localizations, corresponds to Lurie’s scaled unstraightening equivalence. In this ∞–bicategorical context, the relative 2–nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie’s relative nerve when restricted to 1–categories.

Classification : 18D30, 18E35, 18G30, 18G55
Keywords: relative nerve, Grothendieck construction, bicategories

Abellán García, Fernando  1   ; Dyckerhoff, Tobias  1   ; Stern, Walker H  2

1 Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
2 Department of Mathematics, University of Virginia, Charlottesville, VA, United States 
@article{AGT_2020_20_6_a11,
     author = {Abell\'an Garc{\'\i}a, Fernando and Dyckerhoff, Tobias and Stern, Walker H},
     title = {A relative 2{\textendash}nerve},
     journal = {Algebraic and Geometric Topology},
     pages = {3147--3182},
     year = {2020},
     volume = {20},
     number = {6},
     url = {http://geodesic.mathdoc.fr/item/AGT_2020_20_6_a11/}
}
TY  - JOUR
AU  - Abellán García, Fernando
AU  - Dyckerhoff, Tobias
AU  - Stern, Walker H
TI  - A relative 2–nerve
JO  - Algebraic and Geometric Topology
PY  - 2020
SP  - 3147
EP  - 3182
VL  - 20
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/AGT_2020_20_6_a11/
ID  - AGT_2020_20_6_a11
ER  - 
%0 Journal Article
%A Abellán García, Fernando
%A Dyckerhoff, Tobias
%A Stern, Walker H
%T A relative 2–nerve
%J Algebraic and Geometric Topology
%D 2020
%P 3147-3182
%V 20
%N 6
%U http://geodesic.mathdoc.fr/item/AGT_2020_20_6_a11/
%F AGT_2020_20_6_a11
Abellán García, Fernando; Dyckerhoff, Tobias; Stern, Walker H. A relative 2–nerve. Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 3147-3182. http://geodesic.mathdoc.fr/item/AGT_2020_20_6_a11/

[1] D Dugger, D I Spivak, Rigidification of quasi-categories, Algebr. Geom. Topol. 11 (2011) 225 | DOI

[2] T Dyckerhoff, A categorified Dold–Kan correspondence, preprint (2017)

[3] A Grothendieck, Revêtements étales et groupe fondamental (SGA 1), 224, Springer (1971)

[4] Y Harpaz, J Nuiten, M Prasma, Quillen cohomology of (∞,2)–categories, High. Struct. 3 (2019) 17

[5] J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) | DOI

[6] J Lurie, (∞,2)–categories and the Goodwillie calculus, I, preprint (2009)

[7] R Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283 | DOI

[8] D Verity, Weak complicial sets, II : Nerves of complicial Gray-categories, from: "Categories in algebra, geometry and mathematical physics" (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 441 | DOI

[9] D R B Verity, Weak complicial sets, I : Basic homotopy theory, Adv. Math. 219 (2008) 1081 | DOI