This paper holds twofold purposes,

1.

firstly to unfurl a theory of presentations and colimits of enriched monads for subcategories of arities with sufficient generality to accommodate, in case that \(\mathcal{V}\) is locally bounded, the \(\Phi\)-accessible \(\mathcal{V}\)-monads [S. Lack and J. Rosický, Appl. Categ. Struct. 19, No. 1, 363–391 (2011; Zbl 1242.18007)] as well as the \(\mathcal{J}\)-ary \(\mathcal{V}\)-monads for a small and eleutheric system of arities \(\mathcal{J}\hookrightarrow\mathcal{C}\) [R. B. B. Lucyshyn-Wright, Theory Appl. Categ. 31, 101–137 (2016; Zbl 1337.18002)], and

2.

secondly to ensure that the resulting theory of presentation and algebraic colimits covers in full generality such specific settings as the strongly finitary \(\mathcal{V}\)-monads of G. M. Kelly and S. Lack [Appl. Categ. Struct. 1, No. 1, 85–94 (1993; Zbl 0787.18007)], in case that \(\mathcal{V}\) is a complete and cocomplete cartesian closed category or, more generally, a \(\pi\)-category in the sense of F. Borceux and B. Day [J. Pure Appl. Algebra 16, 133–147 (1980; Zbl 0426.18004)], and Wolff’s presentations of \(\mathcal{V}\)-categories by generators and relations for an arbitrary complete and cocomplete \(\mathcal{V}\) [H. Wolff, J. Pure Appl. Algebra 4, 123–135 (1974; Zbl 0282.18010)].

The authors accomplish these objectives by working with enriched monads for a suitable subcategory of arities \(j:\mathcal{J}\hookrightarrow\mathcal{C}\) in a \(\mathcal{V}\)-category \(\mathcal{C}\), where \(\mathcal{V}\) is a complete and cocomplete symmetric monoidal closed category that need not be locally presentable. The results apply when \(\mathcal{C}\) is a locally bounded \(\mathcal{V}\)-category over a locally bounded closed category \(\mathcal{V}\), and in some cases even without these assumptions.
To get these results, the authors make some modest completeness and cocompleteness assumptions on the \(\mathcal{V}\)-category \(\mathcal{C}\) as well as two main assumptions on the subcategory of arities \(j:\mathcal{J} \hookrightarrow\mathcal{C}\).

1.

First, the authors generally assume that \(j:\mathcal{J}\hookrightarrow \mathcal{C}\) is small and eleutheric [R. B. B. Lucyshyn-Wright, Theory Appl. Categ. 31, 101–137 (2016; Zbl 1337.18002)], which is a certain exactness condition guaranteeing that the \(\mathcal{V}\)-endofunctor on \(\mathcal{C}\) that are left Kan extensions along \(j\) are precisely those preserving left Kan extensions along \(j\). They are called \(\mathcal{J}\)-ary \(\mathcal{V}\)-endofunctors.

2.

The authors also assume that \(j:\mathcal{J}\hookrightarrow\mathcal{C} \) abides by a mild boundedness condition, which is defined in terms of certain notions from Kelly’s classical paper [G. M. Kelly, Seminarber. Fachbereich Math., Fernuniv. 6, 5–82 (1980; Zbl 0437.18003); Bull. Aust. Math. Soc. 22, 1–83 (1980; Zbl 0437.18004)] on transfinite constructions in category theory.

The main results on free \(\mathcal{J}\)-ary monads, algebraic colimits of \(\mathcal{J}\)-ary monads, and presentations of \(\mathcal{J}\)-ary monads then hold for any bounded and eleutheric subcategory of arities \(j:\mathcal{J} \hookrightarrow\mathcal{C}\) in a \(\mathcal{V}\)-category \(\mathcal{C}\) abiding by mild assumptions.
The synopsis of the paper goes as follows.

§ 2 and § 3

defines the notion of an eleuthetic subcategory of arities \(j:\mathcal{J}\hookrightarrow\mathcal{C}\) in a \(\mathcal{V}\)-category \(\mathcal{C}\) after reviewing some notation and background

§ 4

defines the notions of \(\mathcal{J}\)-ary \(\mathcal{V}\)-endofunctor and \(\mathcal{J}\)-ary \(\mathcal{V}\)-monad on \(\mathcal{C}\).

§ 5

unfurls the theory of algebraically free monads in the enriched context, generalizing [G. M. Kelly, Seminarber. Fachbereich Math., Fernuniv. 6, 5–82 (1980; Zbl 0437.18003); Bull. Aust. Math. Soc. 22, 1–83 (1980; Zbl 0437.18004)].

§ 6

defines the notion of a bounded subcategory of arities, showing in 6.2.5 and 6.2.6 that if \(j:\mathcal{J}\hookrightarrow\mathcal{C} \) is a bounded subcategory of arities in a cocomplete and cotensored \(\mathcal{V}\)-category \(\mathcal{C}\), then the forgetful functor \[ \mathcal{W}:\boldsymbol{Mnd}_{\mathcal{J}}(\mathcal{C})\rightarrow\boldsymbol{End}_{\mathcal{J}}(\mathcal{C}) \] from \(\mathcal{J}\)-ary \(\mathcal{V}\)-monads on \(\mathcal{C}\) to \(\mathcal{J} \)-ary \(\mathcal{V}\)-endfunctors on \(\mathcal{C}\) is monadic, and that the free \(\mathcal{J}\)-ary \(\mathcal{V}\)-monad on a \(\mathcal{J}\)-ary \(\mathcal{V}\)-endfunctor is algebraically free. These are the first main results in this paper.

§ 7

defines the notion of a \(\Sigma\)-algebra for a -signature \(\Sigma \) in \(\mathcal{C}\), relative to a subcategory of arities \(j:\mathcal{J} \hookrightarrow\mathcal{C}\), showing in 7.7 under certain assumptions that the forgetful functor \[ \boldsymbol{End}_{\mathcal{J}}(\mathcal{C})\rightarrow \boldsymbol{Sig}_{\mathcal{J}}(\mathcal{C}) \] from \(\mathcal{J}\)-ary \(\mathcal{V}\)-endfunctors on \(\mathcal{C}\) to \(\mathcal{J}\)-signatures in \(\mathcal{C}\) is monadic. It is then shown in 7.9 that the forgetful functor \[ \mathcal{U}:\boldsymbol{Mnd}_{\mathcal{J}}(\mathcal{C})\rightarrow\boldsymbol{Sig}_{\mathcal{J}}(\mathcal{C}) \] has a left adjoint, and that the \(\mathcal{V}\)-category of algebras for the free \(\mathcal{J}\)-ary \(\mathcal{V}\)-monad on a \(\mathcal{J}\)-signature \(\Sigma\) is isomorphic to the \(\mathcal{V}\)-category \(\Sigma\) -\(\boldsymbol{Alg}\) of \(\Sigma\)-algebras.

§ 8

establishes by use of S. Lack [J. Pure Appl. Algebra 140, No. 1, 65–73 (1999; Zbl 0974.18005)] in 8.2 that the forgetful functor \[ \mathcal{U}:\boldsymbol{Mnd}_{\mathcal{J}}(\mathcal{C})\rightarrow\boldsymbol{Sig}_{\mathcal{J}}(\mathcal{C}) \] is actually monadic.

§ 9

is concerned with algebraic colimits of \(\mathcal{J}\)-ary \(\mathcal{V}\)-monads, being divided into three subsections. §9.1 proves some results about limits and colimits in limit \(\mathcal{V}\)-categories. §9.2 studies the notion of an algebraic colimi of \(\mathcal{V}\)-monads. §9.3 defines the notion of an algebraic colimit of \(\mathcal{J}\)-ary \(\mathcal{V} \)-monads, demonstrating in 9.3.8 that if \(j:\mathcal{J}\hookrightarrow \mathcal{C}\) is bounded, then the category \(\boldsymbol{Mnd}_{\mathcal{J} }(\mathcal{C})\) of \(\mathcal{J}\)-ary \(\mathcal{V}\)-monads on \(\mathcal{C}\) has small algebraic colimits.

§ 10

defines the notion of a \(\mathcal{J}\)-presentation \(P=(\Sigma,E)\) for a subcategory of arities \(j:\mathcal{J} \hookrightarrow\mathcal{C}\) consisting of \(\mathcal{J}\)-signature morphisms from a \(\mathcal{J}\)-signature \(\Gamma\) (the signature of equations) to the underlying \(\mathcal{J}\)-signature of the free \(\mathcal{J}\)-ary \(\mathcal{V}\)-monad \(\mathbb{T}_{\Sigma}\) on \(\Sigma\). It is shown in 10.1.8 that every \(\mathcal{J}\)-presentation \(P\) presents a \(\mathcal{J}\)-ary \(\mathcal{V}\)-monad \(\mathbb{T}_{P}\), whose \(\mathcal{V}\)-category of algebras turns out in 10.1.8 to be isomorphic to the \(\mathcal{V}\)-category \(P\)-\(\boldsymbol{Alg}\) of \(P\)-algebras for the \(\mathcal{J}\)-presentations \(P=(\Sigma,E)\). It is also shown in 10.1.10 that every \(\mathcal{J}\)-ary \(\mathcal{V}\)-monad has a \(\mathcal{J}\)-presentation.

§ 11

addresses some specimens of \(\mathcal{J}\)-presentations, firstly showing that presentations of \(\mathcal{V}\)-categories by generators and relations are recovered when \(\mathcal{J}\) consists of the representables in a power of \(\mathcal{V}\), while secondly discussing presentations of strongly finitary \(\mathcal{V}\)-monads in cartesian closed topological categories over \(\boldsymbol{Set}\), dealing with internal modules and affine spaces over internal rings (i.e. semirings).

§ 12

summarizes the main results in this paper.