(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
An object in an AB5-category is of finite type if one of the following equivalent conditions hold:
(i) any complete directed set of subobjects of is stationary
(ii) for any complete directed set of subobjects of an object the natural morphism is an isomorphism.
An object is finitely presented if it is of finite type and if for any epimorphism where is of finite type, it follows that is also of finite type. An object in an AB5 category is coherent if it is of finite type and for any morphism where is of finite type is of finite type.
For an exact sequence in an AB5 category the following hold:
For a module over a ring this is equivalent to being finitely generated -module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form where and are finite.
An AB5-category is locally coherent if it has a generating set of coherent objects. If it is such, than every finitely presented object is coherent, and the full subcategory of finitely presented objects is therefore abelian.
Let be a topos.
An object of is called compact if the top element of the poset of subobjects is a compact element.
An object of is called stable if for all morphisms from a compact object , the domain of the kernel pair of is also a compact object.
An object of is called coherent if it is compact and stable.
Let be a small cartesian site, and suppose that is generated by finite covering families. For an object of , let denote the sheaf associated to the presheaf represented by . Then
This is (Johnstone, Theorem D3.3.7).
An object in an (∞,1)-topos is an -coherent object if the slice (∞,1)-topos is an n-coherent (∞,1)-topos
A coherent object which is also n-truncated for some is called a finitely constructible object.
∞Grpd is a coherent (∞,1)-topos and a locally coherent (∞,1)-topos. An object , hence an ∞-groupoid, is an -coherent object if all its homotopy groups in degree are finite. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.
(Lurie SpecSchm, example 3.13)
N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375
Jacob Lurie, section 3 of Spectral Schemes
Ivo Herzog, Contravariant functors on the category of finitely presented modules, Israel J. Math. pdf: The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74(3) (1997), 503-558 pdf
Last revised on July 13, 2017 at 14:37:12. See the history of this page for a list of all contributions to it.