nLab
coherent category

Contents

Definition

A coherent category (also called a pre-logos) is a regular category in which the subobject posets Sub(X)Sub(X) all have finite unions which are preserved by the base change functors f *:Sub(Y)Sub(X)f^*:Sub(Y)\to Sub(X).

A coherent functor serves as a morphism between coherent categories.

The internal logic of a coherent category is coherent logic.

An infinitary coherent category is a well-powered regular category in which the subobject posets Sub(X)Sub(X) have all small unions which are stable under pullback. Infinitary coherent categories are also called geometric categories.

Mike Shulman: I’m starting to feel that well-poweredness shouldn’t be included here. Any thoughts?

See familial regularity and exactness for a general description of the spectrum from regular categories through finitary and infinitary coherent categories.

Properties

Extensivity

Any coherent category automatically has a strict initial object. Moreover, if an object XX is the union of two subobjects AXA\hookrightarrow X and BXB\hookrightarrow X such that AB=0A\cap B = 0, then X=A⨿BX = A\amalg B is their coproduct. Thus, if every pair of objects can be embedded disjointly in some third object, then a coherent category has disjoint finite coproducts and is extensive (or “positive”).

Extensivity is the analogue for coherent categories of exactness for regular categories. A coherent category which is both extensive and exact is called a pretopos.

The coherent topology

Any coherent category CC admits a subcanonical Grothendieck topology in which the covering families are generated by finite, jointly regular-epimorphic families: the coherent coverage. Equivalently, they are generated by single regular epimorphisms and by finite unions of subobjects. 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 RAR\;\rightrightarrows\; A is a kernel pair, then for any f:BAf\colon B\to A, the pullback f *RBf^*R \;\rightrightarrows\; B is also a kernel pair (this is equivalent to the codomain fibration being a stack for the regular topology). 1. If A,BCA,B \rightarrowtail C are a pair of subobjects, then for any f:XAf\colon X\to A and g:YBg\colon Y\to B and any isomorphism f *(AB)g *(AB)f^*(A\cap B) \cong g^*(A\cap B), the pushout

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

exists and is also a pullback.

Topoi of sheaves for coherent topologies on coherent categories are called coherent topoi. (The terminology is slightly confusing, though, because every topos is a coherent category.)

Making categories coherent

Just like the reg/lex completion, there is a “coh/lex completion” which makes an arbitrary finitely complete category into a coherent one in a universal way.

Similarly, there are “pretopos completions” analogous to the ex/reg completion and the ex/lex completion.

Subobjects, slices and internal logic

For any object XX in a coherent category 𝒞\mathcal{C}, the poset of subobjects Sub 𝒞(X)Sub_{\mathcal{C}}(X) is a distributive lattice. For f:XYf : X \to Y any morphism, the base change f *:Sub(Y)Sub(X)f^* : Sub(Y) \to Sub(X) has a left adjoint f:Sub(X)Sub(Y)\exists_f : Sub(X) \to Sub(Y): the dependent sum/existential quantifier along ff.

The corresponding functor/indexed category

Sub:𝒞 opDistLat Sub : \mathcal{C}^{op} \to DistLat

to the category of distributive lattices is the coherent hyperdoctrine of the coherent category 𝒞\mathcal{C}.

Topos of types

See topos of types.

References

Section A1.4 of

Revised on October 25, 2012 22:12:28 by Urs Schreiber (82.169.65.155)