Lie monoids as monoids internal to the category of smooth manifolds?

Question

This question can be thought as a complement to this one.

Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups, seem to deserve a much more complicated definition (see, for instance, 'Lie semigroups and their applications', by Hilgert and Neeb, section 1.4).

Briefly, these are thought as closed subsemigroups of Lie groups, satisfying an extra property. This property, on its turn, is related to the infinitesimal counterpart of the notion of Lie semigroup (in the above reference, the notion of 'Lie wedge', whose definition, consequently, must precede that of a Lie semigroup).

What kind of difficulties appear if one tries to define a Lie monoid simply as a monoid internal to the category of smooth manifolds (or some related category)?

A LITTLE BIT OR FURTHER DISCUSSION

Lie groupoids, on their turn, can be defined as groupoids internal to the category of smooth manifolds. Is there an analogous notion of 'Lie category', in which morphisms are allowed not to be isomorphisms? Of course, the same question holds for its infinitesimal counterpart.

I tried to find some reference dealing with such a notion, but couldn't. Though, it seems to be a reasonable one to consider even within the realm of Lie groupoid theory. For example, if one wants to allow distinct objects to have distinct automorphism groups, but still be connected by morphisms, this notion seems to be a necessary step.

In particular, that's the case if one wants to allow morphisms between distinct objects to be not only isomorphisms between their automorphism groups, but also covering maps between them. I can't think right now of a concrete example coming, say, from Physics, but it sounds possible that the 'internal symmetries' of a system might 'collapse' in this particular way.

Besides that, exactly as Lie groupoids can be considered as natural generalizations of Lie groups (even if this shouldn't be considered the most appropriate point of view, for many reasons...), the 'Lie categories' would be natural generalizations of Lie monoids. Indeed, a 'Lie category' with one object would amount precisely to a Lie monoid.

Any references will be appreciated.

Answer 1

There is indeed a notion of "Lie category", introduced a 1959 paper of Charles Ehresmann: Catégories topologiques et categories différentiables. This is accessible in his OEuvres Complètes, part I, pages 237–250. I can't say it's gained massive traction. Incidentally, I think this is the paper that introduced Lie groupoids, but under the name 'differentiable groupoids'. Most of the 19 citations recorded by MathSciNet seem to be citing it for this purpose. The name of Lie was originally attached to a special case, which you can still see if you read Mackenzie's book on Lie groupoids from the 1980s. (Note that there is a more recent notion of 'differentiable category' in a 2006 paper by Blute–Cockett–Seely that is unrelated, coming from a computer science perspective.)

One nice result is that the core of the underlying category—the maximal subgroupoid—is proved to be a Lie groupoid, though the proof is in a rather old style. This is analogous to the maximal subgroup of the Lie monoid of $n\times n$ matrices with multiplication is a Lie group.