Geodesic
Algebraic categories whose projectives are explicitly free
Theory and applications of categories, Tome 22 (2009), pp. 509-541.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

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.
Classification : 18C20, 05A19, 08B30
Keywords: monads, combinatorics, projective objects, free objects 
@article{TAC_2009_22_a19,
     author = {Mat{\'\i}as Menni},
     title = {Algebraic categories whose projectives are explicitly free},
     journal = {Theory and applications of categories},
     pages = {509--541},
     publisher = {mathdoc},
     volume = {22},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2009_22_a19/}
}
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Matías Menni. Algebraic categories whose projectives are explicitly free. Theory and applications of categories, Tome 22 (2009), pp. 509-541. http://geodesic.mathdoc.fr/item/TAC_2009_22_a19/