Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra of the real line into the quantum structure which is in our case a monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean $\sigma$-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure

Internal Languages of Finitely Complete (∞,1)(\infty, 1)-categories

We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration categories. As a consequence, we deduce a variant of the conjecture asserting that Martin-L\"of Type Theory with dependent sums and intensional identity types is the internal language of $(\infty, 1)$-categories with finite limits.Comment: 41 pages, minor revision

Sharp and fuzzy observables on effect algebras

Observables on effect algebras and their fuzzy versions obtained by means of confidence measures (Markov kernels) are studied. It is shown that, on effect algebras with the (E)-property, given an observable and a confidence measure, there exists a fuzzy version of the observable. Ordering of observables according to their fuzzy properties is introduced, and some minimality conditions with respect to this ordering are found. Applications of some results of classical theory of experiments are considered.Comment: 23 page