nLab coherent object

Definition

In Abelian categories

An object $X$ in an AB5-category $C$ is of finite type if one of the following equivalent conditions hold:

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

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

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

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

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

For a module $M$ over a ring $R$ this is equivalent to $M$ being finitely generated $R$-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form $R^I \to R^J \to M \to 0$ where $I$ and $J$ 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 $C$ be a topos.

Definition

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

Definition

An object $X$ of $C$ is called stable if for all morphisms $Y \to X$ from a compact object $Y$, the domain of the kernel pair $R \rightrightarrows Y$ of $f$ is also a compact object.

Definition

An object $X$ of $C$ is called coherent if it is compact and stable.

Theorem

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

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

This is (Johnstone, Theorem D3.3.7).

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

Definition

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

(Lurie, def. 3.1).

Remark

A coherent object which is also n-truncated for some $n$ 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 $X$, hence an ∞-groupoid, is an $n$-coherent object if all its homotopy groups in degree $k \leq n$ are finite. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.

(Lurie SpecSchm, example 3.13)

See also

• compact object

• coherent topos

• coherent cohomology

• (∞,1)-pretopos

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