Universalities
type: Item
instance of: scholarly article
title: Universalities
zbMATH Open document ID: 0799.18003
DOI: 10.1007/BF00873298
author: Věra Trnková
published in: Applied Categorical Structures
publication date: 1994-08-10T00:00:00Z
review text: Let \({\mathcal V}\) be a cocomplete monoidal category such that \(A \otimes - \) and \(- \otimes A\) preserve colimits for any \({\mathcal V}\)-object \(A\). Then there exist a \({\mathcal V}\)-category \({\mathcal U}\) and a \({\mathcal V}\)-functor \(F : {\mathcal U} \to {\mathcal U}\) such that for every \({\mathcal V}\)-functor \(H : {\mathcal H}_ 1 \to {\mathcal H}_ 2\) between \({\mathcal V}\)-categories \({\mathcal H}_ 1\) and \({\mathcal H}_ 2\) there exist strict \({\mathcal V}\)-isomorphisms \(\varphi_ i : {\mathcal H}_ i \to {\mathcal U}\), \(i = 1,2\), with \(\varphi_ 2 \circ H = F \circ \varphi_ 1\). As a consequence we obtain that there exist a quotient category \({\mathcal U}\) of the category of all topological spaces and all open continuous mappings (or of the category of all metric spaces and all open uniformly continuous mappings, respectively) and a functor \(F : {\mathcal U} \to {\mathcal U}\) such that for every functor \(H : {\mathcal H}_ 1 \to {\mathcal H}_ 2\) between arbitrary categories there exist full embeddings \(\varphi_ i : {\mathcal H}_ i \to {\mathcal U}\), \(i=1,2\), with \(\varphi_ 2 \circ H = F \circ \varphi_ 1\).
Mathematics Subject Classification ID: 18B15
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 617343
zbMATH Keywords: universal category
zbMATH Keywords: \(V\)-category
zbMATH Keywords: monoidal category
reviewed by: Václav Koubek
MaRDI profile type: Publication
cites work: Q4693062
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full work available at URL: https://doi.org/10.1007/bf00873298
OpenAlex ID: W4233261400
description: scientific article; zbMATH DE number 617343