Pseudo-commutativity of KZ 2-monads
The manuscript is advertised by its author as the first step in his attempt to get a rigid algebraic description of the situation, when certain colimits in a category commute with certain limits, the term ``first'' motivated by the fact that the main attention of the current paper is devoted to commutation of colimits with colimits. More precisely, given a complete and cocomplete symmetric monoidal closed \textbf{Cat}-enriched category \(\mathcal{W}\), the paper shows that a specific type of \(\mathcal{W}\)-monads on \(\mathcal{W}\)-categories, the so-called \textit{Kock-Zöberlein} (abbreviated \textit{KZ}) or \textit{lax-idempotent} \(\mathcal{W}\)-monads, are pseudo-commutative, the pseudo-commutativity in question being unique (Theorem~7.3). The main examples of such monads the author is interested in (considered in Section~8) are monads on \(\mathcal{V}\)-\textbf{Cat}, the algebras of which are \(\mathcal{V}\)-categories with a given small class \(\Phi\) of colimits, which essentially is a small class of weights \(\phi:D\rightarrow\mathcal{V}\). On the other hand, there exist pseudo-commutative 2-monads, which are not KZ (Example~7.5), e.g., the 2-monad on \textbf{Cat}, the algebras of which are symmetric strict monoidal categories. The paper starts with a thorough introduction into the main aspects of enriched category theory underlying its developments (Sections~2~-~6), provides its main achievements and their illustrating examples (Sections~7~-~8), and ends with rather routine \(\mathcal{W}\)-enriched versions of certain results concerning categories of algebras w.r.t. a monad (Appendix~A), e.g., the fact that a 2-monad on \(\mathcal{V}\)-\textbf{Cat}, whose algebras are \(\mathcal{V}\)-categories with chosen finite colimits, is finitary. The manuscript is quite technical in its nature, involving a significant number of enriched category-theoretic concepts and trying to provide a brief insight into each of them. The ultimate outcome of such an exposition, however, does not meet the expectations of the author. People, who are not acquainted with the required notions, will be unable to follow the presented results, due to the scarceness of the given theoretical background. On the other hand, those researchers, who do have the necessary knowledge on the topic of the paper, will find the long introduction preceding the actual achievements of the manuscript rather tedious.
scientific article; zbMATH DE number 5969495
scientific article