Using the derived algebraic formalism of Lurie we describe a framework for coalgebraic structures which are neither necessarily commutative nor cocommutative. We show that Thom spectra and homotopical descent data are both natural examples of this kind of structure.

A characterization result for equationally deenable classes of certain coalgebras (including basic hidden algebra) shows that a class of coalgebras is deenable by equations if and only if it is closed under coproducts, quotients, sources of morphisms and representative inclusions. The notions of equation and satisfaction are axiomatized in order to include the diierent approaches in the literat...

This paper gives an overview of recent results concerning the modular derivation of (i) modal specification logics, (ii) notions of simulation together with logical characterisations, and (iii) sound and complete axiomatisations, for systems modelled as coalgebras of functors on Set. Our approach applies directly to an inductivelydefined class of coalgebraic types, which subsumes several types ...

We discuss Osius’s [22] concept of a recursive coalgebra of a functor from the perspective of programming semantics and give some new sufficient conditions for the recursiveness of a functor-coalgebra that are based on comonads, comonad-coalgebras and distributive laws.

We prove that any non L-cocommutative finite Markov L-coalgebra yields at least two solutions of the Yang-Baxter equation and therefore at least two representations of the braid groups. We conclude by a generalisation of these constructions to any coalgebra.

For any L ∞-algebra L we construct an A ∞-algebra structure on the symmetric coalgebra Sym * c (L) and prove that this structure satifies properties generalizing those of the usual universal enveloping algebra. We also obtain an invariant contracting homotopy one the cobar construction of a symmetric coalgebra, by relating it to the combinatorics of permutahedra and semistandard Young tableaux.

In this work we study the induction theory for Hopf group coal-gebra. To reach this goal we define a substructure B of a Hopf group coalgebra H, called subHopf group coalgebra. Also, we introduced the definition of Hopf group suboalgebra and group coisotropic quantum subgroup of H.

We investigate the possibility of deriving metric trace semantics in a coalgebraic framework. First, we generalize a technique for systematically lifting functors from the category Set of sets to the category PMet of pseudometric spaces, by identifying conditions under which also natural transformations, monads and distributive laws can be lifted. By exploiting some recent work on an abstract d...