Research output: Contribution to journalArticlepeer-review


Let M be a monoidal ∞-category with colimits. In this paper we study colimits of M-functors A → B where B is left-tensored over M and A is an M-enriched category. We prove that the enriched Yoneda embedding Y: A → PM(A) yields a universal M-functor. In case when A has a certain monoidal structure, the category of enriched presheaves PM(A) inherits the same monoidal structure and the enriched Yoneda embedding acquires the structure of universal monoidal M-functor.

Original languageEnglish
Pages (from-to)365-422
Number of pages58
JournalTheory and Applications of Categories
Issue number12
StatePublished - 2023

Bibliographical note

Funding Information:
The author was supported by ISF grant 786/19. Received by the editors 2021-02-03 and, in final form, 2023-03-19. Transmitted by Ieke Moerdijk. Published on 2023-03-21. 2020 Mathematics Subject Classification: 18D20, 18N60, 18N70. Key words and phrases: enriched categories, Day convolution, left-tensored categories. © Vladimir Hinich, 2023. Permission to copy for private use granted. 1One can think of colimits for functors of two kinds: functors from one M-enriched category to another, and functors from an M-enriched category to a category left tensored over M. In this paper we deal with this second kind of functors.

Publisher Copyright:
© Vladimir Hinich, 2023.


  • Day convolution
  • enriched categories
  • left-tensored categories

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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