nLab coherent coverage

Definition

For $C$ a coherent category $C$ the coherent coverage on $C$ is the coverage in which the covering families are generated by finite, jointly regular-epimorphic families. Similarly, on an infinitary-coherent category (a.k.a. a “geometric category”), the infinitary-coherent coverage (a.k.a. geometric coverage) is generated by all small jointly regular-epimorphic families.

The corresponding Grothendieck topology is called the coherent topology, making $C$ into a coherent site.

Equivalently, this coverage is generated by single regular epimorphisms and by finite unions of subobjects (resp. small unions in the infinitary case).

Topoi of sheaves for (finitary) coherent topologies on coherent categories are called coherent toposes. (The terminology is slightly confusing, though, because every topos is a coherent category.) Note that every topos is equivalent to a topos of sheaves for the infinitary coherent topology on an infinitary-coherent site, namely itself.

Properties

The coherent coverage is subcanonical.

• If $C$ is extensive, then its coherent topology is generated by the regular topology together with the extensive topology. (In fact, the coherent topology is superextensive.)

• If $C$ is a pretopos, then its self-indexing is a stack for its coherent topology. Exactness and extensivity are stronger than necessary, however; a pair of necessary and sufficient conditions for this to hold are that

  1. If $R\;\rightrightarrows\; A$ is a kernel pair, then for any $f\colon B\to A$, the pullback $f^*R \;\rightrightarrows\; B$ is also a kernel pair (this is equivalent to the codomain fibration being a stack for the regular topology).
  2. If $A,B \rightarrowtail C$ are a pair of subobjects, then for any $f\colon X\to A$ and $g\colon Y\to B$ and any isomorphism $f^*(A\cap B) \cong g^*(A\cap B)$, the pushout
     $$\array{f^*(A\cap B) & \overset{}{\to} & X\\ \downarrow && \downarrow\\ Y& \underset{}{\to} & Z}$$

     exists and is also a pullback.

Related concepts

category	functor	internal logic	theory	hyperdoctrine	subobject poset	coverage	classifying topos
finitely complete category	cartesian functor	cartesian logic	essentially algebraic theory
lextensive category		disjunctive logic
regular category	regular functor	regular logic	regular theory	regular hyperdoctrine	infimum-semilattice	regular coverage	regular topos
coherent category	coherent functor	coherent logic	coherent theory	coherent hyperdoctrine	distributive lattice	coherent coverage	coherent topos
geometric category	geometric functor	geometric logic	geometric theory	geometric hyperdoctrine	frame	geometric coverage	Grothendieck topos
Heyting category	Heyting functor	intuitionistic first-order logic	intuitionistic first-order theory	first-order hyperdoctrine	Heyting algebra
De Morgan Heyting category		intuitionistic first-order logic with weak excluded middle			De Morgan Heyting algebra
Boolean category		classical first-order logic	classical first-order theory	Boolean hyperdoctrine	Boolean algebra
star-autonomous category		multiplicative classical linear logic
symmetric monoidal closed category		multiplicative intuitionistic linear logic
cartesian monoidal category		fragment $\{\&, \top\}$ of linear logic
cocartesian monoidal category		fragment $\{\oplus, 0\}$ of linear logic
cartesian closed category		simply typed lambda calculus

References

Section A1.4 and example C2.1.12 in

• Peter Johnstone, Sketches of an Elephant