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Algebraic categories whose projectives are explicitly free

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Matías Menni

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

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