Re: Where does the term monad come from?

[Zinovy Diskin <zdiskin@gsd.uwaterloo.ca>]

On Fri, Apr 3, 2009 at 12:28 AM, Steve Lack <s.lack@uws.edu.au> wrote:
>
> Finitary monads can also be considered on other base categories than Set,
> especially on locally finitely presentable ones.
>
> It is true that vector spaces are the algebras for a finitary monad on Set.
> There is no need to restrict to finite-dimensional vector spaces; in fact it
> is not true that there is a monad on Set whose algebras are the
> finite-dimensional vector spaces.
>

there is something similar in algebraic logic. The class of locally
finite cylindric/polyadic algebras is not a variety and the forgetful
functor to Set is not monadic (l.f. means that all relations are of
finite arities). In categorical logic (hyperdoctrines), these algebras
are considered in many-sorted signatures, in fact, as algebras over
graphs, and their theory becomes equational (= the corresponding
forgetful functor to Graph is monadic). Probably, it's a general
phenomenon wrt specifying finitary objects: by indexing them with
finite sets (contexts, supports,arities), we get equational theories
over graph-like structures.

In a wider (and partly speculative) setting, the shift from classical
algebraic to categorical logic is a shift from simple signatures and
complex theories to  complex signatures and simple theories. In a
sense, this is what category theory does wherever it applies to
classical problems: it greatly simplifies the logic (and the internal
structure), but pays for this by a complex vocabulary (the external
structure, interface). A typical example is classical vs. categorical
set theories.

Thus, a categorical model is a device with a structurally complex
interface and simple internal logic. An average user prefers, of
course, simple-looking interfaces of classical theories (and
eventually has to pay for this choice but it happens later on...). So,
for marketing categorical models, it's important to provide good
manuals for their complicated interfaces -- what Vaughan just did for
monads.

Zinovy