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National Repository of Grey Literature 2 records found  Search took 0.01 seconds. 
Mapping spaces of algebras over iterated +-construction for polynomial monads
 Grego, Maroš ; Batanin, Michael (advisor) ; Curien, Pierre-Louis (referee)
In a 2019 paper "Polynomial monads and delooping of maping spaces", Batanin and De Leger have introduced an extension of Grothendieck homotopy theory from the cate- gory of small categories to the category of polynomial monads. As an application (among other), they provided a new proof of a famous Tourchin-Dwyer-Hess theorem on explicit double loop space of mapping spaces between the associativity operad and an arbitrary reduced multiplicative operad. In this thesis we generalize Batanin-De Leger results to a sequence of polynomial mon- ads produced by iteration of the Baez and Dolan +-construction (the so called opetopic sequence). For the n-th element of the opetopic sequence, we introduce the monads called k-dimensional bimodules, 0 ≤ k ≤ n which generalize the notions of bimodules and infinitesimal bimodules over the associative operad for non-symmetric operads. The 0-dimensional bimodules are a sequence of categories of opetopes, with each the full subcategory of the next, which generalizes the simplicial category ∆ and the dentroidal category of planar trees Ωp. We show that an explicit double looping of the corresponding mapping space exists for any n ≥ 2, where n = 2 corresponds to the classical case. We provide a further reduceness condition on the multiplicative operad under which the third...
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The incompleteness theorems and Berry's paradox
 Grego, Maroš ; Krajíček, Jan (advisor) ; Kompatscher, Michael (referee)
This thesis is devoted to a formal presentation of an alternative proof of Gödel's first incompleteness theorem, based on the Berry paradox ("the smallest number not definable in under 57 characters", with this definition having less characters and defining this number). The approach used was suggested by an article by G. Chaitin. We define the Kolmogorov complexity of a natural number m as the binary length of the smallest program for the universal Turing machine that on input 0 outputs the number m. Using a formal argument based on the Berry paradox, we show that the property of a (large enough) number n being a lower bound for the Kolmogorov complexity of a number m is not provable in any consistent recursively axiomatizable extension of Robinson arithmetic. But by a counting argument, for all n, it is true for all but finitely many m. This is used to prove the first incompleteness theorem. Another way (by G. S. Boolos) of formalizing the Berry paradox to prove the same theorem is put in the context of the presented approach. 1
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