Some insights on bicategories of fractions: representations and compositions of 2-morphisms

In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory $\mathcal{C}$ together with a suitable class of morphisms $\mathbf{W}$, one can construct a bicategory $\mathcal{C}[\mathbf{W}^{-1}]$, where all the morphisms of $\mathbf{W}$ are turned into internal equivalences, and that is universal with respect to this property. Most of the descriptions leading to this construction were long and heavily based on the axiom of choice. In this paper we considerably simplify the description of the equivalence relation on $2$-morphisms and the constructions of associators, vertical and horizontal compositions in $\mathcal{C}[\mathbf{W}^{-1}]$, thus proving that the axiom of choice is not needed under certain conditions. The simplified description of associators and compositions will also play a crucial role in two forthcoming papers about pseudofunctors and equivalences between bicategories of fractions.Comment: Published in Theory and Applications of Categorie

Some insights on bicategories of fractions - III

We fix any bicategory $\mathscr{A}$ together with a class of morphisms $\mathbf{W}_{\mathscr{A}}$, such that there is a bicategory of fractions $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$. Given another such pair $(\mathscr{B},\mathbf{W}_{\mathscr{B}})$ and any pseudofunctor $\mathcal{F}:\mathscr{A}\rightarrow\mathscr{B}$, we find necessary and sufficient conditions in order to have an induced equivalence of bicategories from $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$ to $\mathscr{B}[\mathbf{W}_{\mathscr{B}}^{-1}]$. In particular, this gives necessary and sufficient conditions in order to have an equivalence from any bicategory of fractions $\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]$ to any given bicategory $\mathscr{B}$.Comment: References updated, some misprints fixe

Synthesis And Characterization Of Polyynes End-Capped By Biphenyl Groups ({\Alpha},{\Omega}-Biphenylpolyynes)

Stable polyyne chains terminated with biphenyl end groups (a,u-biphenylpolyynes) were synthesized in a single step through a simple procedure by using the Cadiot-Chodkiewicz reaction conditions. The a,ubiphenylpolyynes were separated through HPLC analysis and identified by means of their electronic absorption spectra. The a,u-biphenylpolyynes were studied by FT-IR and Raman spectroscopy and the spectral interpretation was supported with DFT calculations. A peculiarly low reactivity of a,u-biphenylpolyynes with ozone was observed.Comment: The research leading to these results has received funding from the European Research Council Consolidator Grant EspLORE (ERC-2016-CoG Grant No.724610