Elimination of quotients in various localisations of premodels into models

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$\Omega$-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allows one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak $(\omega,n)$-categories, weak $\omega$-groupoids and higher moduli stacks.Comment: The text is the same as in v6; this version contains corrections to the published MDPI paper, the main reason for this change is that the diagram of Proposition 3.1 was meant to be a 3 dimensional diagram (while only the front face appeared in the published paper). The wording of some sentences and the diagram of Example 6.42 are changed accordingly. A typo in the table of Ex. 6.42 is correcte

Elimination of Quotients in Various Localisations of Premodels into Models

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/Ω-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allow one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak ω-categories, weak ω-groupoids and higher moduli stacks

Decidability for Non-Standard Conversions in Typed Lambda-Calculi

This thesis studies the decidability of conversions in typed lambda-calculi, along with the algorithms allowing for this decidability. Our study takes in consideration conversions going beyond the traditional beta, eta, or permutative conversions (also called commutative conversions). To decide these conversions, two classes of algorithms compete, the algorithms based on rewriting, here the goal is to decompose and orient the conversion so as to obtain a convergent system, these algorithms then boil down to rewrite the terms until they reach an irreducible forms; and the "reduction free" algorithms where the conversion is decided recursively by a detour via a meta-language. Throughout this thesis, we strive to explain the latter thanks to the former

Fuzzy geometry

The concept of fuzzy space is due independently to Poincaré and Zeeman. (Poincaré used the term "physical continuum", Zeeman the term "tolerance space". I have reluctantly introduced a third expression since my attempts to generate a vocabulary from either of these have all proved impossibly unwieldy.) Both were led to it by the nature of our perception of space, and both adapted to it tools current in topology. Unfortunately, neither examined the application of these tools in complete detail, and as a result the argument from analogy was somewhat over-extended by both. The resemblances to topology are strong; the differences are sometimes glaring and sometimes subtle. In the latter case the difficulties produced by a topologically-conditioned intuition can be severe obstacles to progress. (Certainly, having been reared mathematically as a topologist I have found it necessary to distrust any conclusion whose proof is not painfully precise. ) For this reason many of the proofs in this paper are set out in somewhat more detail than would be natural in a more established field. For this reason also I have here not only set out the positive results I have so far obtained in the subject but, for the benefit of topologists, elaborated on the failures of analogy with topology where a more succinct exposition would have ignored them as dead ends (e.g., in Chap. I, §2)