coherent coverage in nLab

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Definition

For CC a coherent category CC the coherent coverage on CC 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 CC 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 CC 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 CC 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⇉AR\;\rightrightarrows\; A is a kernel pair, then for any f:B→Af\colon B\to A, the pullback f *R⇉Bf^*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↣CA,B \rightarrowtail C are a pair of subobjects, then for any f:X→Af\colon X\to A and g:Y→Bg\colon Y\to B and any isomorphism f *(A∩B)≅g *(A∩B)f^*(A\cap B) \cong g^*(A\cap B), the pushout

     f *(A∩B) → X ↓ ↓ Y → Z\array{f^*(A\cap B) & \overset{}{\to} & X\\ \downarrow && \downarrow\\ Y& \underset{}{\to} & Z}

     exists and is also a pullback.

References

Section A1.4 and example C2.1.12 in

• Peter Johnstone, Sketches of an Elephant