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Register a new userOn functors between categories with colored morphisms
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Yasuhide Numata
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In this paper, we consider categories with colored morphisms and functors such that morphisms assigned to morphisms with a common color have a common color. In this paper, we construct a morphism-colored functor such that any morphism-colored functor from a given small morphism-colored groupoid to any discrete morphism-colored category factors through it. We also apply the main result to a schemoid constructed from a Hamming scheme.
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Each Gr-functor of the type $(varphi,f)$ of a Gr-category of the type $(Pi,C)$ has the obstruction be an element $overline{k}in H^3(Pi,C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type $(varphi,f)$ and the cohomology group $H^2(Pi,C).$ Then the relation of Gr-category theory and the group extension problem can be established and used to prove that each Gr-category is Gr-equivalent to a strict one.
Frobenius monoidal functors preserve duals. We show that conversely, (co)monoidal functors between autonomous categories which preserve duals are Frobenius monoidal. We apply this result to linearly distributive functors between autonomous categories.
The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functionals defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that states on categories as linear functionals defined on category algebras turn out to be generalized of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain representations of category algebras of $^{dagger}$-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs.
For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C,S) of S-spans (s,f) in C with first leg s lying in S, and give an alternative construction of its quotient category C[S^{-1}] of S-fractions. Instead of trying to turn S-morphisms directly into isomorphisms, we turn them separately into retractions and into sections in a universal manner, thus obtaining the quotient categories Retr( C,S) and Sect(C,S). The fraction category C[S^{-1}] is their largest joint quotient category. Without confining S to be a class of monomorphisms of C, we show that Sect(C,S) admits a quotient category, Par(C,S), whose name is justified by two facts. On one hand, for S a class of monomorphisms in C, it returns the category of S-spans in C, also called S-partial maps in this case; on the other hand, we prove that Par(C,S) is a split restriction category (in the sense of Cockett and Lack). A further quotient construction produces even a range category (in the sense of Cockett, Guo and Hofstra), RaPar(C,S), which is still large enough to admit C[S^{-1}] as its quotient. Both, Par and RaPar, are the left adjoints of global 2-adjunctions. When restricting these to their fixed objects, one obtains precisely the 2-equivalences by which their name givers characterized restriction and range categories. Hence, both Par(C,S)$ and RaPar(C,S may be naturally presented as Par(D,T)$ and RaPa(D,T), respectively, where now T is a class of monomorphisms in D. In summary, while there is no {em a priori} need for the exclusive consideration of classes of monomorphisms, one may resort to them naturally
This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give sufficient conditions for a collection of spherical functors to yield a weak representation of the category of tangles, and prove a structure theorem for such representations under certain restrictions.
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