This contribution provides a study of some combinatorial monoids, namely finite decomposition and locally finite monoids, using some tools from category theory. One corrects the lack of functoriality of the construction of the large algebras of finite decomposition monoids by considering the later as monoid objects of the category of sets with finite-fiber maps. Moreover it is proved that an algebraic monoid (i.e., a commutative bialgebra) may be associated to any finite decomposition monoid, and that locally finite monoids furthermore provide algebraic groups (i.e., commutative Hopf algebras), by attaching in the first case a monoid scheme to the large algebra, and in the second case a group scheme to a subgroup of invertible elements in the large algebra.

Beginning with a general concept of bilinear maps and pairings in any (symmetric) monoidal closed category, we provide a description of the moduli space of pairings, i.e., a complete solution to the classification problem of pairings (which is exactly the same as the classification of finite abelian groups up to isomorphism), up to isomorphism, from finite abelian groups to the group of complex roots of unity. We also provide an algebraic description of the monoid of equivalence classes of pairings as a projective limit of monoids (with a zero). Moreover we prove a geometric property satisfied by this monoid, namely that it embeds as a submonoid of the rational points of an algebraic monoid (on an algebraic closed field).

A pair of finite abelian groups is called ``pairing admissible'' if there exists a non-degenerate bilinear map, called a ``pairing'', from the product of these groups to a third one. Using some elementary group-theory, as a main result is proved that two abelian groups are pairing admissible if, and only if, the canonical bilinear map to their tensor product is itself a pairing, if and only if, they share the same exponent. One also observes that being pairing admissible is an equivalence relation among the class of all finite abelian groups, and one proves that the corresponding quotient set admits a structure of a semilattice isomorphic to that of positive integers under divisibility.

The symbolic method, introduced by P. Flajolet and R. Sedgewick to compute (ordinary, exponential, and multivariate) generating functions in a systematic way, makes use of basic objects and some construction rules -- which, for certain of them, are shown to be functorial in the present contribution -- to combine them in order to obtain more complex objects of the same kind (combinatorial classes) so that their generating functions are computed in a way (actually also functorial) that reflects these combinations. In this paper, in which we adopt a functorial point of view, is provided a more general framework to study these objects, allowing transfinite cardinalities. We prove that the functional (actually functorial) association of a class to its generating function is a universal (Grothendieck) invariant. Finally we also present some constructions of (combinatorial or not) classes that extend the known ones by application of universal algebra.