The Grothendieck construction for model categories

The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As a main result, we establish an equivalence between suitable diagrams of model categories indexed by $\mathcal{M}$ and a new notion of \textbf{model fibrations} over $\mathcal{M}$. When $\mathcal{M}$ is a model category, our construction endows the Grothendieck construction with a model structure which gives a presentation of Lurie's $\infty$-categorical Grothendieck construction and enjoys several good formal properties. We apply our construction to various examples, yielding model structures on strict and weak group actions and on modules over algebra objects in suitable monoidal model categories.Comment: Includes revisions based on the comments of the refere

On the fibration method for zero-cycles and rational points

Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Th\'el\`ene, Sansuc, Kato and Saito in the 1980's. We prove that these conjectures are compatible with fibrations, for fibrations into rationally connected varieties over a curve. In particular, they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers. We prove the analogous result for rational points, conditionally on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.Comment: 54 pages; v3: minor updates, added Remark 9.12(ii), v4: improved exposition, final versio

Equation of Motion of an Electric Charge

The appearance of the time derivative of the acceleration in the equation of motion (EOM) of an electric charge is studied. It is shown that when an electric charge is accelerated, a stress force exists in the curved electric field of the accelerated charge, and this force is proportional to the acceleration. This stress force acts as a reaction force which is responsible for the creation of the radiation (instead of the "radiation reaction force" that actually does not exist at low velocities). Thus the initial acceleration should be supplied as an initial condition for the solution of the EOM of an electric charge. It is also shown that in certain cases, like periodic motions of an electric charge, the term that includes the time derivative of the acceleration, represents the stress reaction force.Comment: 12 pages, 2 figure