Unicity of enrichment over \(\mathcal Cat\) or \(\mathcal Gpd\)
type: Item
instance of: scholarly article
title: Unicity of enrichment over \(\mathcal Cat\) or \(\mathcal Gpd\)
zbMATH Open document ID: 1225.18007
published in: Applied Categorical Structures
publication date: 2011-03-30T00:00:00Z
review text: The paper is motivated by the question posed by \textit{G.~M.~Kelly} on whether a monad on \(\mathcal Cat\) (the category of small categories) can be enriched over \(\mathcal Cat\) in a unique way. The author shows a sufficient condition (the existence of tensors or cotensors with the arrow category) for an ordinary monad on a 2-category to have at most one enrichment (\(\mathcal Cat\) can be easily replaced with the category \(\mathcal Gpd\) of groupoids). On the other hand, there exist monads on \(\mathcal Cat\), which have no enrichment and, moreover, enrichment over the categories different from \(\mathcal Cat\) and \(\mathcal Gpd\) can be not unique (the respective examples are provided in the manuscript).
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18C15
zbMATH DE Number: 5871640
zbMATH Keywords: 2-category
zbMATH Keywords: cotensors
zbMATH Keywords: enrichment
zbMATH Keywords: monad
zbMATH Keywords: tensors
author: John Power
reviewed by: Sergejs Solovjovs
MaRDI profile type: Publication
full work available at URL: s10485-009-9202-7
OpenAlex ID: W2039988913
cites work: Two-dimensional monad theory
cites work: Algebraic categories with few monoidal biclosed structures or none
cites work: Enrichment through variation
cites work: Elementary observations on 2-categorical limits
cites work: Q3934579
cites work: Q4380422
cites work: Why tricategories?
cites work: A Unified Category-theoretic Semantics for Binding Signatures in Substructural Logics
DOI: 10.1007/S10485-009-9202-7
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description: scientific article; zbMATH DE number 5871640