12,829 research outputs found
Extended deformation functors
We introduce a precise notion, in terms of few Schlessinger's type
conditions, of extended deformation functors which is compatible with most of
recent ideas in the Derived Deformation Theory (DDT) program and with geometric
examples. With this notion we develop the (extended) analogue of Schlessinger
and obstruction theories. The inverse mapping theorem holds for natural
transformations of extended deformation functors and all such functors with
finite dimensional tangent space are prorepresentable in the homotopy category.Comment: Contains the previously announced part I
Coherence for Categorified Operadic Theories
It has long been known that every weak monoidal category A is equivalent via
monoidal functors and monoidal natural transformations to a strict monoidal
category st(A). We generalise the definition of weak monoidal category to give
a definition of weak P-category for any strongly regular (operadic) theory P,
and show that every weak P-category is equivalent via P-functors and
P-transformations to a strict P-category. This strictification functor is then
shown to have an interesting universal property.Comment: 13 pages, 1 figure. Presented at 82nd PSSL, Glasgow, May 200
Quantum Genetics, Quantum Automata and Quantum Computation
The concepts of quantum automata and quantum computation are studied in the context of quantum genetics and genetic networks with nonlinear dynamics. In a previous publication (Baianu,1971a) the formal concept of quantum automaton was introduced and its possible implications for genetic and metabolic activities in living cells and organisms were considered. This was followed by a report on quantum and abstract, symbolic computation based on the theory of categories, functors and natural transformations (Baianu,1971b). The notions of topological semigroup, quantum automaton,or quantum computer, were then suggested with a view to their potential applications to the analogous simulation of biological systems, and especially genetic activities and nonlinear dynamics in genetic networks. Further, detailed studies of nonlinear dynamics in genetic networks were carried out in categories of n-valued, Lukasiewicz Logic Algebras that showed significant dissimilarities (Baianu, 1977) from Bolean models of human neural networks (McCullough and Pitts,1945). Molecular models in terms of categories, functors and natural transformations were then formulated for uni-molecular chemical transformations, multi-molecular chemical and biochemical transformations (Baianu, 1983,2004a). Previous applications of computer modeling, classical automata theory, and relational biology to molecular biology, oncogenesis and medicine were extensively reviewed and several important conclusions were reached regarding both the potential and limitations of the computation-assisted modeling of biological systems, and especially complex organisms such as Homo sapiens sapiens(Baianu,1987). Novel approaches to solving the realization problems of Relational Biology models in Complex System Biology are introduced in terms of natural transformations between functors of such molecular categories. Several applications of such natural transformations of functors were then presented to protein biosynthesis, embryogenesis and nuclear transplant experiments. Other possible realizations in Molecular Biology and Relational Biology of Organisms are here suggested in terms of quantum automata models of Quantum Genetics and Interactomics. Future developments of this novel approach are likely to also include: Fuzzy Relations in Biology and Epigenomics, Relational Biology modeling of Complex Immunological and Hormonal regulatory systems, n-categories and Topoi of Lukasiewicz Logic Algebras and Intuitionistic Logic (Heyting) Algebras for modeling nonlinear dynamics and cognitive processes in complex neural networks that are present in the human brain, as well as stochastic modeling of genetic networks in Lukasiewicz Logic Algebras
Obvious natural morphisms of sheaves are unique
We prove that a large class of natural transformations (consisting roughly of
those constructed via composition from the "functorial" or "base change"
transformations) between two functors of the form
actually has only one element, and thus that any diagram of such maps
necessarily commutes. We identify the precise axioms defining what we call a
"geofibered category" that ensure that such a coherence theorem exists. Our
results apply to all the usual sheaf-theoretic contexts of algebraic geometry.
The analogous result that would include any other of the six functors remains
unknown.Comment: 52 pages. Final draft, version accepted to TA
Grothendieck categories as a bilocalization of linear sites
We prove that the 2-category Grt of Grothendieck abelian categories with
colimit preserving functors and natural transformations is a bicategory of
fractions in the sense of Pronk of the 2-category Site of linear sites with
continuous morphisms of sites and natural transformations. This result can
potentially be used to make the tensor product of Grothendieck categories from
earlier work by Lowen, Shoikhet and the author into a bi-monoidal structure on
Grt
Failure of Brown representability in derived categories
Let T be a triangulated category with coproducts, C the full subcategory of
compact objects in T. If T is the homotopy category of spectra, Adams proved
the following in [Adams71]: All contravariant homological functors C --> Ab are
the restrictions of representable functors on T, and all natural
transformations are the restrictions of morphisms in T.
It has been something of a mystery, to what extent this generalises to other
triangulated categories. In [Neeman97], it was proved that Adams' theorem
remains true as long as C is countable, but can fail in general. The failure
exhibited was that there can be natural transformations not arising from maps
in T. A puzzling open problem remained: Is every homological functor the
restriction of a representable functor on T? In a recent paper, Beligiannis
made some progress. But in this article, we settle the problem. The answer is
no. There are examples of derived categories T = D(R) of rings, and
contravariant homological functors C --> Ab which are not restrictions of
representables.Comment: 22 pages, to appear in Topology. http://jdc.math.uwo.ca Lots of minor
revisions. This version should closely match the published versio
Natural Transformations of Organismic Structures
The mathematical structures underlying the theories of organismic sets, (M, R)-systems and molecular sets are shown to be transformed naturally within the theory of categories and functors. Their natural transformations allow the comparison of distinct entities, as well as the modelling of dynamics in “organismic” structures
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