Algebraic Categories Whose Projectives Are Explicitly Free
نویسنده
چکیده
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 (MA0,m), but it is also the case that the object A0 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.
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