The various approaches to categories of internal coalgebras in a cocomplete category $\mathcal{C}$ and a variety in particular, as discussed for example by Kan, Freyd and Bergman, are described in terms of Lawvere theories $\mathcal{T}$. We prove that these categories $\mathbf{Coalg}(\mathcal{T},\mathcal{C})$ are locally presentable and have a left adjoint forgetful functor into the base category $\mathcal{C}$, provided that the latter is locally presentable; this generalizes and strengthens Bergman's recent completeness result for categories of internal coalgebras in varieties. Under the additional assumption that finite sums of monomorphisms are monomorphisms these categories are covarieties in the sense of Ad\'amek-Porst. It will moreover be shown that Freyd's canonical constructions of internal coalgebras in a variety are left adjoints. In particular, every commutative variety $\mathcal{V}$ can be coreflectively embedded into the category$\mathbf{Coalg}(\mathcal{T},\mathcal{C})$ of internal coalgebras, if $\mathcal{V}$ admits a morphism of Lawvere theories from $\mathcal{T}$ to the theory of $\mathcal{V}$. Special instances of these coreflectors are the group-like elements and the primitive elements functors, known from Hopf algebra theory.

The modern mathematical foundations of quantum physics deal with operator algebras. The purpose of this contribution is an analysis of a new kind of these algebras, namely semigroup objects in the category of Hilbert spaces, hereafter referred to as "Hilbert algebras", using Banach (co)algebraic techniques, mainly the Gelfand transform. In particular, it is shown that any commutative Hilbert algebra splits as an orthogonal sum of a closed sub-coalgebra spanned by the group-like elements and its radical, and that the underlying Banach coalgebra of a Hilbert algebra embeds into a Banach coalgebra of continuous functions on a compact semigroup. These results are based on the monoidal functorial relations between Hilbert spaces, with the Hilbertian tensor product, and Banach spaces, with either the injective or the projective tensor products, and also on the fact that any Hilbert algebra has an adjoint Hilbert coalgebra which shares the same underlying Hilbert space.

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.

One of the most general setting in which linear algebra and matrix calculus is still possible is that of linear categories. This means that we can manipulate matrices with entries in a base linear category rather than a base ring. In this contribution we develop the foundations of this matrix calculus with a particular emphasis on the question of matrix representability of linear operators. In particular are provided sufficient conditions under which a linear operator may be represented faithfully by a matrix in this generalized setting. Our results specialized to the classical ones when is considered a linear category with only one object, i.e., an algebra.

Varyadic function symbols are used to represent functions with a variable arity such as flattened terms in the context of an associative theory. Polymorphism is another way to deal with variability not as the number of operands but as their sorts. In this contribution we show that the usual approaches of mono and many-sorted algebras suffices to handle and simulate these two situations. In the literature the construction of the free algebra over a varyadic signature is often neglected, being qualified as the "usual one" (by reference to the same construction in the context of a fixed-arity signature). We show that actually we must proceed with more caution to do this. Finally we use our relation between polymorphic terms and many-sorted algebras to give an actual freeness status to a part of a construction of J. Meseguer in Conditional rewriting logic as unified model of concurrency, Theoretical Computer Science, volume 96, pages 73-155, 1992.

It is well-known that any $\Sigma$-algebra may be represented as an algebra whose operator domain consists in constants and a single binary operation: this holds according to the usual currying technique that transforms a function that takes multiple arguments into several applications of one-variable functions. In this contribution we show that a similar result may be obtained by using a structure of a so-called Cantor or Jónsson-Tarski algebra. Both constructions give rise to functors (representations) to the category of $\Sigma$-algebras which admit a left adjoint, and the free $\Sigma$-algebra on a set is shown to be faithfully represented by derived operators in signatures consisting of functions of arity less or equal than $2$.

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.