Sentence examples for category of sections from inspiring English sources
6.1, we associate a special prefibration over (Delta ) to a comonad and construct a functor from the category of coalgebras over a comonad to the category of sections of the prefibration (this is Lemma 6.4).
end{aligned} (9.1)The functor is of course not an equivalence, and the category on the right-hand side is not triangulated it is only the category of sections of a prefibration with triangulated fibers.
Moreover, even if we forget the additional structure, the corresponding fibered category (mathcal {C}rightarrow I) itself is not a DG category, and neither is the category of sections ({text {Sec}}(mathcal {C})).
Thus a collection of categories (mathcal {C}_i), (i=0,1,2) and functors (Phi _{ij}:mathcal {C}_i rightarrow C_j), (0 le i category of sections of this precobiration has nothing to do with the iterated comma-category (mathsf{R}(Phi _cdot )).
Corresponding approaches fall under the third category of Section 4.1: further developments or generalizations of quantum theory.
Producing model structures on categories of sections of Grothendieck prefibrations is usually a highly non-trivial exercise.
Since we work over simplicial replacements anyway, Reedy structures are a natural tool to use, and all we need to do is to generalize them to the categories of sections of a Grothendieck prefibration.
If I and (I') are small, (a^*) induces a functor on the categories of sections, and by the universal property of cartesian liftings, we have a natural morphism begin{aligned} alpha (a):gamma _0^* rightarrow a^* circ gamma _1^* end{aligned} (4.17)of functors from ({text {Sec}}(I,mathcal {C})) to ({text {Sec}}(I',gamma _0^*mathcal {C})).
Passing to the categories of sections, we obtain a functor begin{aligned} widetilde{m}^*:{text {Sec}}((M(M i)),widetilde{t}^*rho ^*mathcal {C}') rightarrow mathcal {C}_i^{'M(M i))}, end{aligned}and if we equip its target with the projective model structure, then the functor is right-derivable.
Another corollary of Proposition 8.4 is extended functoriality for the categories of derived sections.
Thus we have a functor (a^*:mathcal {C}rightarrow sigma _dagger ^*sigma ^*mathcal {C}), this functor is an equivalence by Definition 5.8 (i), and this equivalence induces an equivalence between the categories of special sections.
