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

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 - II

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 pseudofunctor $\mathcal{G}:\mathscr{A}[\mathbf{W}_{\mathscr{A}}^{-1}]\rightarrow \mathscr{B}[\mathbf{W}_{\mathscr{B}}^{-1}]$. Moreover, we give a simple description of $\mathcal{G}$ in the case when the class $\mathbf{W}_{\mathscr{B}}$ is "right saturated".Comment: References updated, some details changed in the proofs of section

A bicategory of reduced orbifolds from the point of view of differential geometry - I

We describe a bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we first define a $2$-category $(\mathcal{R}ed\,\mathcal{A}tl)$ whose objects are reduced orbifold atlases (on any paracompact, second countable, Hausdorff topological space). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl, while the definitions of $2$-morphisms and compositions of them is new in this setup. Using the bicalculus of fractions described by D. Pronk, we are able to construct the bicategory $(\mathcal{R}ed\,\mathcal{O}rb)$ from the $2$-category $(\mathcal{R}ed\,\mathcal{A}tl)$. We prove that $(\mathcal{R}ed\,\mathcal{O}rb)$ is equivalent to the bicategory of reduced orbifolds described in terms of proper, effective, \'etale Lie groupoids by D. Pronk and I. Moerdijk and to the $2$-category of reduced orbifolds described by several authors in the past in terms of a suitable class of differentiable Deligne-Mumford stacks.Comment: An essential part of the proof has been generalized and is now contained in a series of separated papers about bicategories of fractions. The description of the homotopy category has also been moved to a separated pape

A bicategory of reduced orbifolds from the point of view of differential geometry - I

We describe a bicategory (Red Orb) of reduced orbifolds in the framework of differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we first define a 2-category (Red Atl) whose objects are reduced orbifold atlases (on paracompact, second countable, Hausdorff topological spaces). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl, while the definition of 2-morphisms and compositions of them is new in this setup. Using the bicalculus of fractions described by D. Pronk, we are able to construct from such a 2-category the bicategory (Red Orb). We prove that it is equivalent to the bicategory of reduced orbifolds described in terms of proper, effective, étale Lie groupoids by D. Pronk and I. Moerdijk and to the 2-category of reduced orbifolds described by several authors in the past in terms of a suitable class of differentiable Deligne-Mumford stacks.

Theoretical investigation and computational evaluation of overtone and combination features in resonance Raman spectra of polyenes and carotenoids

We review the theory for overtones and combinations in resonant Raman spectroscopy introduced by Nafie, Stein and Peticolas in 1971 on the basis of time-ordered diagrams, and we apply it to β-carotene with the support of density functional theory calculations. Comparison with experimental results obtained by Tasumi’s group in 1994 is provided. The theory here presented allows a prompt evaluation of resonant Raman intensities with presently available quantum chemistry tools