Semantic Factorization and Descent

Let $\mathbb{A}$ be a $2$-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism $p$ exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the higher cokernel of $p$ is up to isomorphism the same as the semantic factorization of $p$, either one existing if the other does. The result can be seen as a counterpart account to the celebrated B\'{e}nabou-Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of $p$ trivially hold whenever $p$ has a left adjoint and, hence, in this case, we find monadicity to be a $2$-dimensional exact condition on $p$, namely, to be an effective faithful morphism of the $2$-category $\mathbb{A}$.Comment: Full revision, new diagrams, 48 page

The spin structure of the proton at low xx and low Q2Q^2 in two-dimensional bins from COMPASS

The longitudinal double spin asymmetries $A_1^p$ and the spin dependent structure function of the proton $g_1^p$ were extracted from COMPASS data in the region of low Bjorken scaling variable $x$ and low photon virtuality $Q^2$. The data were taken in 2007 and 2011 from scattering of polarised muons off polarised protons, resulting in a sample that is 150 times larger than the one from the previous experiment SMC that pioneered studies in this kinematic region. For the first time, $A_1^p$ and $g_1^p$ were evaluated in this region in two-dimensional bins of kinematic variables: $(x,Q^2)$, $(\nu ,Q^2)$, $(x,\nu)$ and $(Q^2,x)$. The following kinematic region was investigated: $4\times 10^{-5}<x<4\times 10^{-2}$, $0.001$~(GeV/$c$)$^2<Q^2<1$~(GeV/$c$)$^2$ and $14$~GeV$<\nu <194$~GeV. The obtained results were confronted with theoretical models.Comment: 5 pages, 3 figures, DIS 2016 Conference Proceedings, to appear in the proceedings of the XXIV International Workshop on Deep-Inelastic Scattering and Related Subjects, DESY Hamburg, Germany, April 11-15, 201