I was about to write a long answer, but varkor stole the scene. I think his answer is pretty good and I'd like to complement it with a bunch of references that I find relevant.

The general mood of the references below is that a monad over Set is, in particular, an enriched monad over Set. This is not irrelevant when it comes to the interpretation of monads as theories. Of course, in the spirit of nervous monads (and relative monads in general), one can still develop a lot the theory. But when we take a more Yoneda-point of view, say in the spirit of a completeness-like theorem, enrichment starts to matter.

• Rosický, Metric monads.
• Rosický, Discrete equational theories.
• Adámek et al, Quantitative Algebras and a Classification of Metric Monads.
• Adámek, Varieties of quantitative algebras as categories.

The papers above were all inspired by recent developments initiated by Plotkin et al on the general topic of quantitative algebras, but (especially Rosický's work) cover a lot of other cases providing great intuition.

I should stress that Quantitative Algebras and a Classification of Metric Monads stresses on the gap between enriched monads and general monads, clarifying (or I should say hinting) the difference in the two approaches when it comes to logical repercussions.

There is also a nice talk by Adámek on youtube where he discusses this issue.

I was about to write a long answer, but varkor stole the scene. I think his answer is pretty good and I'd like to complement it with a bunch of references that I find relevant.

The general mood of the references below is that a monad over Set is, in particular, an enriched monad over Set. This is not irrelevant when it comes to the interpretation of monads as theories. Of course, in the spirit of nervous monads (and relative monads in general), one can still develop a lot the theory. But when we take a more Yoneda-point of view, say in the spirit of a completeness-like theorem, enrichment starts to matter.

• Rosický, Metric monads.
• Rosický, Discrete equational theories.
• Adámek et al, Quantitative Algebras and a Classification of Metric Monads.
• Adámek, Varieties of quantitative algebras as categories.

The papers above were all inspired by recent developments initiated by Plotkin et al on the general topic of quantitative algebras, but (especially Rosický's work) cover a lot of other cases providing great intuition.

I should stress that Quantitative Algebras and a Classification of Metric Monads stresses on the gap between enriched monads and general monads, clarifying (or I should say hinting) the difference in the two approaches when it comes to logical repercussions.

There is also a nice talk by Adámek on youtube given at the Brno Algebra Seminar where he discusses this issue.

I was about to write a long answer, but varkor stole the scene. I think his answer is pretty good and I'd like to complement it with a bunch of references that I find relevant.

The general mood of the references below is that a monad over Set is, in particular, an enriched monad over Set. This is not irrelevant when it comes to the interpretation of monads as theories. Of course, in the spirit of nervous monads (and relative monads in general), one can still develop a lot the theory. But when we take a more Yoneda-point of view, say in the spirit of a completeness-like theorem, enrichment starts to matter.

• Rosicky, Metric monads.
• Rosicky, Discrete equational theories.
• Adamek et al, Quantitative Algebras and a Classification of Metric Monads
• Adamek, Varieties of quantitative algebras as categories

The papers above were all inspired by recent developments initiated by Plotkin et al on the general topic of quantitative algebras, but (especially Rosicky's work) cover a lot of other cases providing great intuition.

I should stress that Quantitative Algebras and a Classification of Metric Monads stresses on the gap between enriched monads and general monads, clarifying (or I should say hinting) the difference in the two approaches when it comes to logical repercussions.

There is also a nice talk by Adamek on youtube where he discusses this issue.

I was about to write a long answer, but varkor stole the scene. I think his answer is pretty good and I'd like to complement it with a bunch of references that I find relevant.

The general mood of the references below is that a monad over Set is, in particular, an enriched monad over Set. This is not irrelevant when it comes to the interpretation of monads as theories. Of course, in the spirit of nervous monads (and relative monads in general), one can still develop a lot the theory. But when we take a more Yoneda-point of view, say in the spirit of a completeness-like theorem, enrichment starts to matter.

• Rosický, Metric monads.
• Rosický, Discrete equational theories.
• Adámek et al, Quantitative Algebras and a Classification of Metric Monads.
• Adámek, Varieties of quantitative algebras as categories.

The papers above were all inspired by recent developments initiated by Plotkin et al on the general topic of quantitative algebras, but (especially Rosický's work) cover a lot of other cases providing great intuition.

I should stress that Quantitative Algebras and a Classification of Metric Monads stresses on the gap between enriched monads and general monads, clarifying (or I should say hinting) the difference in the two approaches when it comes to logical repercussions.

There is also a nice talk by Adámek on youtube where he discusses this issue.