Enriched algebraic theories and monads for a system of arities (Q2808141)

There exists several approaches to categorical algebra, motivated by the idea of universal algebra to study algebraic structures in terms of operations and equations. In particular, \textit{F. W. Lawvere} [Repr. Theory Appl. Categ. 2004, No. 5, 1--121 (2004; Zbl 1062.18004)] introduced the notion of an algebraic theory, which provides a categorical representation of algebras with finitary operations. More precisely, an algebraic theory \(\mathcal T\) is a denumerable set \(\{T^0,T^1,\dots,T^n,\dots\}\) of distinct objects, where each object \(T^n\) is the \(n\)-th power of the object \(T=T^1\). A \(\mathcal T\)-algebra is a functor \(F:\mathcal T\rightarrow\mathcal Set\), which preserves finite products. A homomorphism of \(\mathcal T\)-algebras is a natural transformation. A modification of this setting of \textit{F. E. J. Linton} [in: Proc. Conf. Categor. Algebra, La Jolla 1965, 84--94 (1966; Zbl 0201.35003)] (natural numbers \(n\) are replaced with arbitrary sets \(J\)) allows one to capture infinitary operations as well. Later on, \textit{F. Borceux} and \textit{B. Day} [J. Pure Appl. Algebra 16, 133--147 (1980; Zbl 0426.18004)] proposed to replace the category \(\mathcal Set\) with a symmetric monoidal closed category \(\mathcal V\) and take \(\mathcal T\) to be a \(\mathcal V\)-enriched category. Moreover, \textit{J. Power} [Theory Appl. Categ. 6, 83--93 (1999; Zbl 0943.18003)] suggested to take the operation arities \(J\) to be finitely presentable objects of \(\mathcal V\) (\(\mathcal V\) is then assumed to be locally finitely presentable as a closed category). The present paper introduces a general notion of enriched algebraic theory, which includes all of the above examples. In particular, a system of arities is understood to be a fully faithful strong symmetric monoidal \(\mathcal V\)-functor \(j:\mathcal J\rightarrowtail\mathcal V\). The author shows modest conditions on a \(\mathcal J\)-theory \(\mathcal T\), which imply the existence of the \(\mathcal V\)-category \(\mathcal T\)-\(\mathcal Alg\) of \(\mathcal T\)-algebras and its monadicity over \(\mathcal V\) ( Theorem 8.9 on page~123). He also establishes an equivalence between \(\mathcal J\)-theories and \(\mathcal J\)-ary \(\mathcal V\)-monads on \(\mathcal V\), i.e., those \(\mathcal V\)-monads, which preserve left Kan extensions along \(j\) (Theorem 11.8 on page 130). Lastly, he provides a characterization theorem for \(\mathcal J\)-algebraic \(\mathcal V\)-categories over \(\mathcal V\), i.e., characterizes those \(\mathcal V\)-functors \(\mathcal A\rightarrow\mathcal V\), which are equivalent to the forgetful \(\mathcal V\)-functor \(\mathcal T\)-\(\mathcal Alg\rightarrow\mathcal V\) for some \(\mathcal J\)-theory \(\mathcal T\) (Theorem 12.2 on page 134).NEWLINENEWLINEThe paper is well written, provides most of its required preliminaries, and will be of interest to all categorical algebraists.