nLab coherent category

Definition

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

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 regular category in which the subobject posets $Sub(X)$ have all small unions which are stable under pullback. Infinitary coherent categories are also called geometric categories. (These are sometimes also required to be well-powered; see their page for details.)

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

Examples

• Set is a coherent category.

• More generally, any topos is coherent and any pretopos is coherent.

• Even more generally, any locally cartesian closed category with finite colimits is coherent, and thus any quasitopos is coherent.

• Every Heyting category is a coherent category.

• If $C$ is coherent, so is the functor category $C^D$ for any $D$.

• A slice category of a coherent category is coherent.

• Any distributive lattice, regarded as a thin category, is coherent.

• The category of PERs over a PCA is coherent.

Properties

Extensivity

Any coherent category automatically has a strict initial object. Moreover, if an object $X$ is the union of two subobjects $A\hookrightarrow X$ and $B\hookrightarrow X$ such that $A\cap B = 0$, then $X = A\amalg B$ 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 $C$ 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 $C$ 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.

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

The corresponding functor/indexed category

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

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

Topos of types

See topos of types.

Related concepts

• 2-category version: coherent 2-category

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 of

• Peter Johnstone, Sketches of an Elephant