Sentence examples for the derived category from inspiring English sources
The corresponding triangulated category is the derived category ({mathcal {D}}_c(X)) of complexes with coherent homology, and it is in fact equivalent to the derived category ({mathcal {D}}({text {Coh}}(X))).
In fact, one can use model structures already to define the derived category ({mathcal {D}}(mathcal {C})) of an abelian category (mathcal {C}).
Analogously, the k-linear stable model pair (langle C^{pf}_cdot ({text {QCoh}}(X)),C_cdot ({text {QCoh}}(X))rangle ) produces the derived category ({mathcal {D}}^{pf}(X)) of perfect complexes on X.
The formalism of derived categories that appeared in [46] internalizes this independence: to define the derived category ({mathcal {D}}(mathcal {C})), one takes the category (C_cdot (mathcal {C})) of chain complexes of objects in (mathcal {C}), and formally inverts quasiisomorphisms.
On the other hand, the same triangulated category ({mathcal {D}}) can appear as the derived category of different abelian ones (if needed, these can be located inside ({mathcal {D}}) by the machinery of t-structures of [4]).
For triangulated categories, the statement is wrong, and moreover, one cannot recover the derived category ({mathcal {D}}({text {Sec}}(mathcal {C}))) from the family of the derived categories ({mathcal {D}}(mathcal {C}_i)), (i in I).
The derived categories indicated that it may be valuable to have the possibility in choosing from different types of CTs.
The derived categories were: very active vigorous activity, vigorous activity, occasional vigorous activity, light activity, and no activity.
The derived categories were sorted into three overarching themes: collaboration between HCP and informal carers, collaboration among HCP and the impact of resources and healthcare system.
The Greek derived category names include the now widely used aposematic and cryptic, alongside such epithets as "apatetic", "procryptic", "anticryptic", "episematic" and "epigamic", with variants.
In this case, ({text {DSec}}(Delta ^o,{text {Shv}}(X_{an},k))) contains as a full subcategory the equivariant derived category ({text {Shv}}_G(Y_{an},k)) of [5].
