∞-ary regular and exact categories
arity class: unary, finitary, infinitary
regularity
regular category = unary regular
coherent category = finitary regular
geometric category = infinitary regular
exactness
exact category = unary exact
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.
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.
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.
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
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).
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
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.)
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.
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
to the category of distributive lattices is the coherent hyperdoctrine of the coherent category $\mathcal{C}$.
See topos of types.
Section A1.4 of
Last revised on November 16, 2022 at 06:01:24. See the history of this page for a list of all contributions to it.