From: Carsten Fuhrmann <C.Fuhrmann@cs.bham.ac.uk>
To: categories@mta.ca
Subject: Re: Kleisli and colimits
Date: Wed, 25 Apr 2001 17:04:34 +0100 [thread overview]
Message-ID: <3AE6F592.C9298FD8@cs.bham.ac.uk> (raw)
In-Reply-To: <E14sQJw-00034c-00@plover.dpmms.cam.ac.uk>
The construction of the Kleisli category can be turned into a
left adjoint functor whose domain is the category of Mnd monads
(on arbitrary categories) and monad morphisms. However, the
codomain of this functor is not Cat, but a category AbsKl whose
objects I call "abstract Kleisli categories". An abstract
Kleisli category is a category K together with a functor L:K->K,
a natural transformation \epsilon: L->Id, and a (not generally
natural) transformation \theta:Id->L, such that
(1) \theta_L is a natural transformation
(2) L\theta o \theta = \theta_L o \theta
(3) \epsilon o \theta = id
(3) L\epsilon o \theta_L = id
A morphism K->K' of AbsKl is a functor that preserves the
solutions of the non-naturality square
\theta o f = Lf o \theta (I)
The Kleisli construction forms a functor Mnd->AbsKl. Its right
adjoint sends an abstract Kleisli category K to the evident monad
on the subcategory given by the solutions of Equation (I). The
counit of this adjunction is in fact an iso, so AbsKl is a full
reflective subcategory of Mnd. The full subcategory of Mnd which
is equivalent to Abskl is given by those monads for which every
component of the unit is a regular mono.
Cheers,
Carsten
prev parent reply other threads:[~2001-04-25 16:04 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2001-04-25 14:31 Tom Leinster
2001-04-25 16:04 ` Carsten Fuhrmann [this message]
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