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coherent object

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(,1)(\infty,1)-Topos Theory

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Definition

In Abelian categories

An object XX in an AB5-category? CC is of finite type if one of the following equivalent conditions hold:

(i) any complete directed set {X i} iI\{X_i\}_{i\in I} of subobjects of XX is stationary

(ii) for any complete directed set {Y i} iI\{Y_i\}_{i\in I} of subobjects of an object YY the natural morphism colim iC(X,Y i)C(X,Y)\operatorname{colim}_i C(X,Y_i) \to C(X,Y) is an isomorphism.

An object XX is finitely presented if it is of finite type and if for any epimorphism p:YXp : Y \to X where YY is of finite type, it follows that kerp\operatorname{ker} p is also of finite type. An object XX in an AB5 category is coherent if it is of finite type and for any morphism f:YXf : Y \to X where YY is of finite type kerf\operatorname{ker} f is of finite type.

For an exact sequence 0XXX00 \to X' \to X \to X'' \to 0 in an AB5 category the following hold:

  1. if XX' and XX'' are finitely presented, then XX is finitely presented;
  2. if XX is finitely presented and XX' of finite type, then XX'' is finitely presented;
  3. if XX is coherent and XX' of finite type then XX'' is also coherent.

For a module MM over a ring RR this is equivalent to MM being finitely generated RR-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form R IR JM0R^I \to R^J \to M \to 0 where II and JJ 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.

In 1-topos theory

Let CC be a topos.

Definition

An object XX of CC is called compact if the top element of the poset of subobjects Sub(X)Sub(X) is a compact element.

Definition

An object XX of CC is called stable if for all morphisms YXY \to X from a compact object YY, the domain of the kernel pair RYR \rightrightarrows Y of ff is also a compact object.

Definition

An object XX of CC is called coherent if it is compact and stable.

Theorem

Let (C,τ)(C, \tau) be a small cartesian site, and suppose that τ\tau is generated by finite covering families. For an object XX of CC, let l(X)l(X) denote the sheaf associated to the presheaf represented by XX. Then

  • l(X)l(X) is a coherent object of the topos Sh(C,τ)Sh(C, \tau), for all objects XX in CC,
  • if (C,τ)(C, \tau) is further a pretopos with its coherent coverage, then every coherent object of Sh(C,τ)Sh(C, \tau) is isomorphic to l(X)l(X) for some XX.

This is (Johnstone, Theorem D3.3.7).

In (,1)(\infty,1)-topos theory

Definition

An object XX in an (∞,1)-topos H\mathbf{H} is an nn-coherent object if the slice (∞,1)-topos H /X\mathbf{H}_{/X} is an n-coherent (∞,1)-topos

(Lurie, def. 3.1).

Remark

A coherent object which is also n-truncated for some nn is called a finitely constructible object.

(Lurie pAdic, def. 2.3.1)

Examples

Example

∞Grpd is a coherent (∞,1)-topos and a locally coherent (∞,1)-topos. An object XX, hence an ∞-groupoid, is an nn-coherent object if all its homotopy groups in degree knk \leq n are finite. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.

(Lurie SpecSchm, example 3.13)

See also

References

  • N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

  • Peter Johnstone, Sketches of an elephant, D3.3.

  • 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 10:37:12. See the history of this page for a list of all contributions to it.