[[!include topos theory - contents]]
[[!include (infinity,1)-topos - contents]]
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:
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.
Let $C$ be a topos.
An object $X$ of $C$ is called compact if the top element of the poset of subobjects $Sub(X)$ is a compact element.
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.
An object $X$ of $C$ is called coherent if it is compact and stable.
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
This is (Johnstone, Theorem D3.3.7).
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
A coherent object which is also n-truncated for some $n$ is called a finitely constructible object.
∞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)
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 10:37:12. See the history of this page for a list of all contributions to it.