TY - JOUR
T1 - Yoneda lemma for enriched ∞-categories
AU - Hinich, Vladimir
N1 - Funding Information: A part of this work was done during the author's stay at Radboud University at Nijmegen, Utrecht University, MPIM and UC Berkeley. The author is very grateful to these institutions for the excellent working conditions. Numerous discussions with Ieke Moerdijk were very helpful. John Francis' advice for a proof of Proposition 3.6.7 is appreciated. We are very grateful to R. Haugseng who pointed out an error in the first version of the manuscript. We are also very grateful to the anonymous referee who has made an incredible job trying to make the manuscript better. The work was supported by ISF 446/15 and 786/19 grants. Funding Information: A part of this work was done during the author's stay at Radboud University at Nijmegen, Utrecht University, MPIM and UC Berkeley. The author is very grateful to these institutions for the excellent working conditions. Numerous discussions with Ieke Moerdijk were very helpful. John Francis' advice for a proof of Proposition 3.6.7 is appreciated. We are very grateful to R. Haugseng who pointed out an error in the first version of the manuscript. We are also very grateful to the anonymous referee who has made an incredible job trying to make the manuscript better. The work was supported by ISF 446/15 and 786/19 grants. Publisher Copyright: © 2020
PY - 2020/6/24
Y1 - 2020/6/24
N2 - We continue the study of enriched ∞-categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched ∞-categories are associative monoids in an especially designed monoidal category of enriched quivers. We prove that, in the case where the monoidal structure in the basic category M comes from the direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. We apply the Yoneda lemma to the study of correspondences of enriched (for instance, higher) ∞-categories.
AB - We continue the study of enriched ∞-categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched ∞-categories are associative monoids in an especially designed monoidal category of enriched quivers. We prove that, in the case where the monoidal structure in the basic category M comes from the direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. We apply the Yoneda lemma to the study of correspondences of enriched (for instance, higher) ∞-categories.
KW - Enriched infinity categories
KW - Yoneda lemma
UR - http://www.scopus.com/inward/record.url?scp=85082590738&partnerID=8YFLogxK
U2 - https://doi.org/10.1016/j.aim.2020.107129
DO - https://doi.org/10.1016/j.aim.2020.107129
M3 - Article
SN - 0001-8708
VL - 367
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 107129
ER -