Exploring the boundaries of monad tensorability on Set (Q2848367)

Before reviewing the article under consideration, let us introduce, following [\textit{E. G. Manes}, Algebraic theories. New York-Heidelberg-Berlin: Springer-Verlag (1976; Zbl 0353.18007)], the following notation. Let \(\mathbf{S}\) and \(\mathbf{T}\) be monads (\(\equiv\) algebraic theories \(\equiv\) triples) on \(\mathbf{Set}\). The category of \(\mathbf{S}\)-\(\mathbf{T}\) bialgebras and homomorphisms of \(\mathbf{S}\)-\(\mathbf{T}\) bialgebras is denoted by \(\mathbf{Set}^{\mathbf{S}\otimes\mathbf{T}}\) and the forgetful functor from \(\mathbf{Set}^{\mathbf{S}\otimes\mathbf{T}}\) to \(\mathbf{Set}\) by \(U^{\mathbf{S}\otimes\mathbf{T}}\).NEWLINENEWLINEAs is well known, tensor products of monads were considered, among others, by [\textit{P. Freyd}, Colloq. Math. 14, 89--106 (1966; Zbl 0144.01003); \textit{E. G. Manes}, A triple miscellany: some aspects of the theory of algebras over a triple. Middletown, CT: Wesleyan University (Diss.) (1967); Zbl 0353.18007; \textit{F. W. Lawvere}, Lect. Notes Math. 61, 41--61 (1968; Zbl 0204.33802); \textit{J. R. Isbell}, Am. J. Math. 94, 535--596 (1972; Zbl 0439.18009); \textit{J. L. Freire Nistal} and \textit{M. A. López López}, Rev. Mat. Univ. Complutense Madr. 6, No. 1, 13--25 (1993; Zbl 0805.18004)], where the work by Isbell is, unfortunately, not as well known as it really deserved to be. NEWLINENEWLINENEWLINE NEWLINELet us recall that the tensor product of two monads \(\mathbf{S}\) and \(\mathbf{T}\) on \(\mathbf{Set}\) exists as long as the forgetful functor \(U^{\mathbf{S}\otimes\mathbf{T}}: \mathbf{Set}^{\mathbf{S}\otimes\mathbf{T}}\rightarrow \mathbf{Set}\) is monadic (\(\equiv\) algebraic). Isbell proved (in 3.11 of [Zbl 0439.18009]) that \(\mathbf{S}\otimes\mathbf{T}\) does not exist if \(\mathbf{S}\) is the theory whose algebras are vector spaces (for some field) and \(\mathbf{T}\) is suitably chosen. Concretely, Isbell states and proves in [Zbl 0439.18009] that there is a varietal theory \(\mathbf{T}\) such that the category of vector spaces (over any field) in \(\mathbf{T}\)-algebras is not varietal. For this he assumes, for convenience, that the field is countable, that all vector spaces are infinite-dimensional and that \(\mathbf{T}\) has a \(0\)-ary operation \(0\), a unary operation \(\sigma\), and \(m\)-ary \(w_{m}\) for each small cardinal \(m\geq\aleph_{0}\) (and composites of these). We recall that, for Isbell in [Zbl 0439.18009], a cardinal \(m\) is small if it is smaller than an arbitrary, but fixed, strongly inaccessible cardinal. The laws say that \(w_{m}\) vanishes on each \(m\)-tuple whose terms lie in a subalgebra generated by fewer than \(m\) elements (by listing all such \(m\)-tuples formed from \(n<m\) variables). NEWLINENEWLINENEWLINE NEWLINEIn the article under review, the authors prove that the tensor product of two monads need not in general exist by presenting two counterexamples. The first counterexample involves the finite powerset monad and an intricated unranked monad \(\mathbf{M}\). We point out that the authors consider that the underlying signature of the theory \(\mathcal{T}_{\mathbf{M}}\) -- associated to the unranked monad \(\mathbf{M}\) -- has, in particular, for every \(\kappa\), a constant \(c\), and that the theory has as underlying system of equations: \(f_{\alpha,\beta,i}(t_{\alpha',\beta',i'}\mid (\alpha',\beta',i')\in B_{\alpha}) = c\), whenever the \(t_{\alpha',\beta',i'}\) are terms in variables drawn from some set \(X\) of cardinality strictly smaller that that of \(\alpha\) (the \(f_{\alpha,\beta,i}\) are operation symbols of arity \(B_{\alpha}\) for each \((\alpha,\beta,i)\in B_{\kappa+1}\), \(\alpha>0\), where, for every ordinal \(\delta\), \(B_{\delta}\) is the set of all triples \((\alpha,\beta,i)\) such that \(\alpha<\delta\), \(\beta<\alpha^{+}\), and \(i<\omega\)). The second counterexample was stated by \textit{S. Goncharov} and \textit{L. Schröder} in [Lect. Notes Comput. Sci. 6859, 208--221 (2011; Zbl 1287.18006)]. On the other hand, the authors prove that for every regular cardinal \(\kappa>\omega\), the monad \(\mathbf{P}^{\star}_{\kappa}\) (whose underlying functor \(\mathrm{P}^{\star}_{\kappa}\) assigns to a set \(X\) the set \(\mathrm{P}^{\star}_{\kappa}(X)\) of all nonempty subsets of \(X\) of cardinality less than \(\kappa\)) and \(\mathbf{P}_{\kappa}\) (whose underlying functor \(\mathrm{P}_{\kappa}\) assigns to a set \(X\) the set \(\mathrm{P}_{\kappa}(X)\) of all subsets of \(X\) of cardinality less than \(\kappa\)) are genuinely tensorable, i.e., such that, for every monad \(\mathbf{S}\) in \(\mathbf{Set}\), \(\mathbf{P}^{\star}_{\kappa}\otimes \mathbf{S}\) and \(\mathbf{P}_{\kappa}\otimes \mathbf{S}\) exist and have small free algebras.