The effective model structure and $\infty $-groupoid objects
Forum of Mathematics, Sigma, Tome 10 (2022)
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For a category $\mathcal {E}$ with finite limits and well-behaved countable coproducts, we construct a model structure, called the effective model structure, on the category of simplicial objects in $\mathcal {E}$, generalising the Kan–Quillen model structure on simplicial sets. We then prove that the effective model structure is left and right proper and satisfies descent in the sense of Rezk. As a consequence, we obtain that the associated $\infty $-category has finite limits, colimits satisfying descent, and is locally Cartesian closed when $\mathcal {E}$ is but is not a higher topos in general. We also characterise the $\infty $-category presented by the effective model structure, showing that it is the full sub-category of presheaves on $\mathcal {E}$ spanned by Kan complexes in $\mathcal {E}$, a result that suggests a close analogy with the theory of exact completions.
@article{10_1017_fms_2022_13,
author = {Nicola Gambino and Simon Henry and Christian Sattler and Karol Szumi{\l}o},
title = {The effective model structure and $\infty $-groupoid objects},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {10},
year = {2022},
doi = {10.1017/fms.2022.13},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.13/}
}
TY - JOUR AU - Nicola Gambino AU - Simon Henry AU - Christian Sattler AU - Karol Szumiło TI - The effective model structure and $\infty $-groupoid objects JO - Forum of Mathematics, Sigma PY - 2022 VL - 10 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.13/ DO - 10.1017/fms.2022.13 LA - en ID - 10_1017_fms_2022_13 ER -
%0 Journal Article %A Nicola Gambino %A Simon Henry %A Christian Sattler %A Karol Szumiło %T The effective model structure and $\infty $-groupoid objects %J Forum of Mathematics, Sigma %D 2022 %V 10 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.13/ %R 10.1017/fms.2022.13 %G en %F 10_1017_fms_2022_13
Nicola Gambino; Simon Henry; Christian Sattler; Karol Szumiło. The effective model structure and $\infty $-groupoid objects. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.13
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