Sentence examples for categories of sections from inspiring English sources

As long as a work fits within one of the general subject matter categories of sections 102 and 103, the bill prevents the States from protecting it even if it fails to achieve Federal statutory copyright because it is too minimal or lacking in originality to qualify, or because it has fallen into the public domain.

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.

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.

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})).

Paul Lewis, Marc Tsurumaki, and David J. Lewis have developed seven categories of section, revealed in structures ranging from simple one-story buildings to complex structures featuring stacked forms, fantastical shapes, internal holes, inclines, sheared planes, nested forms, or combinations thereof.

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).

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})).

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.

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.