#
${\cal M}$-Completeness is seldom monadic over graphs

##
Jiri Adamek and G. M. Kelly

For a set ${\cal M}$ of graphs the category ${\bf Cat}_{\cal M}$ of all
${\cal M}$-complete categories and all strictly ${\cal M}$-continuous
functors is known to be monadic over ${\bf Cat}$. The question of
monadicity of ${\bf Cat}_{\cal M}$ over the category of graphs is known to
have an affirmative answer when ${\cal M}$ specifies either (i) all finite
limits, or (ii) all finite products, or (iii) equalizers and terminal
objects, or (iv) just terminal objects. We prove that, conversely, these
four cases are (essentially) the only cases of monadicity of $\Cat_\M$
over the category of graphs, provided that ${\cal M}$ is a set of finite
graphs containing the empty graph.

Keywords: category, graph, limit, adjunction.

2000 MSC: 18A35, 18A10, 18C15.

*Theory and Applications of Categories*, Vol. 7, 2000, No. 8, pp 171-205.

http://www.tac.mta.ca/tac/volumes/7/n8/n8.dvi

http://www.tac.mta.ca/tac/volumes/7/n8/n8.ps

http://www.tac.mta.ca/tac/volumes/7/n8/n8.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/7/n8/n8.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/7/n8/n8.ps

TAC Home