Geodesic
${\cal M}$-Completeness is seldom monadic over graphs
Theory and applications of categories, Tome 7 (2000), pp. 171-205.

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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.
Classification : 18A35, 18A10, 18C15.
Keywords: category, graph, limit, adjunction. 
@article{TAC_2000_7_a7,
     author = {Jiri Adamek and G. M. Kelly},
     title = {${\cal M}${-Completeness} is seldom monadic over graphs},
     journal = {Theory and applications of categories},
     pages = {171--205},
     publisher = {mathdoc},
     volume = {7},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2000_7_a7/}
}
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Jiri Adamek; G. M. Kelly. ${\cal M}$-Completeness is seldom monadic over graphs. Theory and applications of categories, Tome 7 (2000), pp. 171-205. http://geodesic.mathdoc.fr/item/TAC_2000_7_a7/