Monoids in functor categories [was: Re: Where does the term monad come from?]

Monoids in functor categories [was: Re: Where does the term monad come from?]

[Fred E.J. Linton]

On Sun, 12 Apr 2009 09:38:18 AM EDT, Thorsten Altenkirch 
<txa@Cs.Nott.AC.UK>, writing to Steve Lack <s.lack@uws.edu.au> 
and <categories@mta.ca>, asked, inter alia: 

> > ... [snip] ...
> 
> ... My question was rather wether
> people have studied monoids in categories of functors which are not
> endofunctors. ... [snip again] ...
> 

Yes, they have. Two quick examples of such categories, 
and what monoids in them boil down to:

1) simplicial sets (cf. Gabriel-Zisman or D.M. Kan for
just how this is a functor category) -- here monoids 
are "simplicial monoids." Of particular interest, 
of course: simplicial groups.

2) modules over a fixed (commutative, say) ring R (a
functor category of the form [R, Ab] from the one-object
additive category R to the category Ab of abelian groups,
consisting of course only of the additive functors) --
here monoids boil down to R-algebras (what in the older
van der Waerden terminology were called hypercomplex
systems over R).

In each instance, of course, one must be careful to
specify correctly just which "product" bifunctor on
the functor category is to be used in the definition
of "monoid" -- in case (2) it's not the usual 
(cartesian) product but, rather, a suitable tensor 
product one that one wants to be using.

I'll let others provide other illustrative examples.

Cheers, -- Fred