Let M = (M, m, u) be a monad and let (MX, m) be the free M-algebra on the object X. Consider an M-algebra (A, a), a retraction r : (MX, m) --> (A, a) and a section t : (A, a) --> (MX, m) of r. The retract (A, a) is not free in general. We observe that for many monads with a `combinatorial flavor' such a retract is not only a free algebra (MA_0, m), but it is also the case that the object A_0 of generators is determined in a canonical way by the section t. We give a precise form of this property, prove a characterization, and discuss examples from combinatorics, universal algebra, convexity and topos theory.
Keywords: monads, combinatorics, projective objects, free objects
2000 MSC: 18C20, 05A19, 08B30
Theory and Applications of Categories,
Vol. 22, 2009,
No. 20, pp 509-541.
http://www.tac.mta.ca/tac/volumes/22/20/22-20.dvi
http://www.tac.mta.ca/tac/volumes/22/20/22-20.ps
http://www.tac.mta.ca/tac/volumes/22/20/22-20.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/20/22-20.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/20/22-20.ps
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/22/20/22-20.pdf