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Authors
Marek Zawadowski
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We characterize the category of co-semi-analytic functors and describe an action of semi-analytic functors on co-semi-analytic functors.
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We describe an abstract 2-categorical setting to study various notions of polynomial and analytic functors and monads.
This paper has been withdrawn and replaced by arXiv:1309.5035.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.
In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the established presentation of such structures as algebras for monads on presheaf categories, we describe a characteristic property of the associated monads---the shapeliness of the title---which says that any two operations of the same shape agree. An important part of this work is the study of analytic functors between presheaf categories, which are a common generalisation of Joyals analytic endofunctors on sets and of the parametric right adjoint functors on presheaf categories introduced by Diers and studied by Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among the analytic endofunctors, and may be characterised as the submonads of a universal analytic monad with exactly one operation of each shape. In fact, shapeliness also gives a way to define the data and axioms of a structure directly from its graphical calculus, by generating a free shapely monad on the basic operations of the calculus. In this paper we do this for some of the examples listed above; in future work, we intend to do so for graphical calculi such as Milners bigraphs, Lafonts interaction nets, or Girards multiplicative proof nets, thereby obtaining canonical notions of denotational model.
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.
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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