We study the existence of universal measuring comonoids P(A,B) for a pair of monoids A, B in a braided monoidal closed category, and the associated enrichment of the category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if A is a bimonoid and B is a commutative monoid, then P(A,B) is a bimonoid; in addition, if A is a cocommutative Hopf monoid then P(A,B) always is Hopf. If A is a Hopf monoid, not necessarily cocommutative, then P(A,B) is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative description of the dualizable P(A,B)‐comodules and use the theory of Hopf (co)monads. We explore the examples of universal measuring comonoids in vector spaces and graded spaces.

A. A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as 'machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.

Particulate matter from biomass burning emissions affects air quality, ecosystems and climate; however, quantifying these effects requires that the connection between primary emissions and secondary aerosol production is firmly established....