From: "Tom Leinster" <t.leinster@maths.gla.ac.uk>
To: categories@mta.ca
Subject: Monads on finite categories
Date: Sat, 25 Mar 2006 21:32:47 -0000 (GMT) [thread overview]
Message-ID: <E1FNS0H-0004x6-VP@mailserv.mta.ca> (raw)
Dear All,
1. Are there significant or interesting examples of monads on finite
categories? I want to look beyond monads on posets, a.k.a. closure
operators. (Since a finite category with binary sums or products
is necessarily a poset, some of the usual examples of monads reduce
to this case.) I can only think of one class of examples
(described below), and I don't know if it's particularly
significant.
2. Any monad on a finite category is idempotent. Is this widely
known?
Thanks.
Tom
* * *
The class of examples: let A be a finite Cauchy-complete category.
Let M be the 2-element monoid consisting of the identity and an
idempotent, so that [M, A] is the category of idempotents in A. Then
the diagonal functor A ---> [M, A] has adjoints on both sides. The
induced monad on A is trivial, but that on [M, A] is not. (It sends
an idempotent e to 1_a, where a is the object through which e splits.)
next reply other threads:[~2006-03-25 21:32 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2006-03-25 21:32 Tom Leinster [this message]
2006-03-27 8:06 Reinhard Boerger
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