Subobject classifier

Abstract: A topos is a category which looks and behaves very much like the category of sets, and so it may be thought of as a universe for mathematical discourses. One of the very useful topoi in many branches of mathematics as well as in computer sciences is the topos MSet, of sets with an action of a monoid M on them. It is well known that MSet, being isomorphic to the functor category Set M , is a topos. Here, we explicitly give the ingredients of a topos in MSet and investigate their properties for the working scientists and computer scientists. Among other things, we give some equivalent conditions, such as the left Ore condition, to Ω, the subobject classifier of MSet, being a Stone algera. Also the free and the cofree objects, as well as, limits and colimits are discussed in MSet.

Abstract: A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology.

Abstract: Homotopy Type Theory may be seen as an internal language for the $\infty$-category of weak $\infty$-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak $\infty$-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak $\infty$-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those `discrete' groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of `elementary' $\infty$-toposes. We prove that sets in homotopy type theory form a $\Pi W$-pretopos. This is similar to the fact that the $0$-truncation of an $\infty$-topos is a topos. We show that both a subobject classifier and a $0$-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover, the $0$-object classifier for sets is a function between $1$-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.