| Title:
|
Quillen cohomology of ($\infty,2$)-categories (English) |
| Author:
|
Harpaz, Yonatan |
| Author:
|
Nuiten, Joost |
| Author:
|
Prasma, Matan |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
3 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
17-66 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study the homotopy theory of parameterized spectrum objects in the $\infty$-category of $(\infty,2)$-categories, as well as the Quillen cohomology of an $(\infty,2)$-category with coefficients in such a parameterized spectrum. More precisely, we construct an analogue of the twisted arrow category for an $(\infty,2)$-category $\Bbb C$, which we call its twisted 2-cell $\infty$-category. We then establish an equivalence between parameterized spectrum objects over $\Bbb C$, and diagrams of spectra indexed by the twisted 2-cell $\infty$-category of $\Bbb C$. Under this equivalence, the Quillen cohomology of $\Bbb C$ with values in such a diagram of spectra is identified with the two-fold suspension of its inverse limit spectrum. As an application, we provide an alternative, obstruction-theoretic proof of the fact that adjunctions between $(\infty,1)$-categories are uniquely determined at the level of the homotopy (3,2)-category of Cat$_\infty$. (English) |
| Keyword:
|
Quillen cohomology |
| Keyword:
|
($\infty,2$)-category |
| Keyword:
|
tangent category |
| Keyword:
|
spectrum |
| Keyword:
|
Grothendieck construction |
| MSC:
|
18D05 |
| MSC:
|
55P42 |
| MSC:
|
55S35 |
| MSC:
|
55T25 |
| idZBL:
|
Zbl 1418.55005 |
| idMR:
|
MR3939045 |
| DOI:
|
10.21136/HS.2019.02 |
| . |
| Date available:
|
2026-03-10T18:56:49Z |
| Last updated:
|
2026-03-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153409 |
| . |
| Reference:
|
[1] Buckley, M.: Fibred 2-categories and bicategories.Journal of Pure and Applied Algebra, 218.6, 2014, p. 1034–1074 10.1016/j.jpaa.2013.11.002 |
| Reference:
|
[2] Dwyer, W. G., Kan, D. M., Smith, J. H.: An obstruction theory for simplicial categories.Indagationes Mathematicae, 89.2, North-Holland |
| Reference:
|
[3] Francis, J.: The tangent complex and Hochschild cohomology of ℰ_(n)-rings.Compositio Mathematica, 149.3, p. 430–480 10.1112/S0010437X12000140 |
| Reference:
|
[4] Harpaz, Y.: Gray products and lax limits - a primer on (∞,2)-categories.in preparation |
| Reference:
|
[5] Harpaz, Y., Nuiten, J., Prasma, M.: The tangent bundle of a model category.preprint arxiv:1802.08031 http://arxiv.org/pdf/1802.08031 |
| Reference:
|
[6] Harpaz, Y., Nuiten, J., Prasma, M.: Tangent categories of algebras over operads.preprint arxiv:1612.02607 http://arxiv.org/pdf/1612.02607 |
| Reference:
|
[7] Harpaz, Y., Nuiten, J., Prasma, M.: The abstract cotangent complex and Quillen cohomology of enriched categories.Journal of Topology, 11.3, p. 752–798 10.1112/topo.12074 |
| Reference:
|
[8] Harpaz, Y., Nuiten, J., Prasma, M.: The Postnikov tower of higher categories.in preparation |
| Reference:
|
[9] Harpaz, Y., Prasma, M.: The Grothendieck construction for model categories.Advances in Mathematics, 281, 2015, p. 1306-1363 10.1016/j.aim.2015.03.031 |
| Reference:
|
[10] Heller, A.: Stable homotopy theories and stabilization.Journal of Pure and Applied Algebra, 115.2, p. 113–130 10.1016/S0022-4049(96)00116-8 |
| Reference:
|
[11] Hirschhorn, P. S.: Model categories and their localizations.Mathematical Surveys and Monographs, 99. American Mathematical Society, Providence, RI, xvi+457 pp. Zbl 1017.55001 |
| Reference:
|
[12] Hovey, M.: Spectra and symmetric spectra in general model categories.Journal of Pure and Applied Algebra, 165.1, p. 63–127 10.1016/S0022-4049(00)00172-9 |
| Reference:
|
[13] Lurie, J.: Stable infinity categories.preprint arxiv:math/0608228 http://arxiv.org/pdf/math/0608228 |
| Reference:
|
[14] Lurie, J.: Higher topos theory.No. 170. Princeton University Press |
| Reference:
|
[15] Lurie, J.: (Infinity, 2)-Categories and the Goodwillie Calculus I.available at: http://www.math.harvard.edu/ lurie/papers/GoodwillieI.pdf |
| Reference:
|
[16] Lurie, J.: On the classification of topological field theories.available at http://www.math.harvard.edu/ lurie/papers/cobordism.pdf |
| Reference:
|
[17] Lurie, J.: Higher Algebra.available at http://www.math.harvard.edu/ lurie/papers/higheralgebra.pdf |
| Reference:
|
[18] Maltsiniotis, G.: La théorie de l’homotopie de Grothendieck.Société mathématique de France |
| Reference:
|
[19] Nguyen, H. K.: Obstruction theory for higher categories.PhD thesis, in preparation |
| Reference:
|
[20] Quillen, D.: Homotopical algebra.Vol. 43, Lecture Notes in Mathematics |
| Reference:
|
[21] Riehl, E., Verity, D.: Homotopy coherent adjunctions and the formal theory of monads.Advances in Mathematics, 286, p. 802–888 |
| Reference:
|
[22] Street, R.: Limits indexed by category-valued 2-functors.Journal of Pure and Applied Algebra, 8.2, p. 149–181 |
| . |
