nLab κ-ary site

Idea

A $\kappa$-ary site is a site whose covering sieves are determined by $\kappa$-small covering families, and which has a very weak sort of finite limits. These conditions get weaker as $\kappa$ gets larger, until when $\kappa$ is the size of the universe, every small site is $\kappa$-ary.

$\kappa$-ary sites are a very general (perhaps the most general) appropriate input for $\kappa$-ary exact completion.

Definition

Let $\kappa$ be an arity class.

Definition

A site $C$ is weakly $\kappa$-ary if for any covering sieve $R$ of an object $V$ in $C$, there exists a $\kappa$-small family $\{p_i: U_i \to V\}_i$ in $C$ such that (1) each $p_i \in R$, and (2) the sieve generated by $\{p_i\}$ is a covering sieve.

This definition can also be rephrased purely in terms of the covering families; see (Shulman).

Definition

Let $C$ be a site and $G:D\to C$ a functor. A local $\kappa$-prelimit of $G$ is a $\kappa$-small family of cones $\{q_i: \Delta L \to G \}_i$ in $C$ such that for any cone $r:\Delta u \to G$, the sieve $\{ p: v\to u | r p$ factors through some $q_i \}$ is a covering sieve of $u$.

Definition

A $\kappa$-ary site is a weakly $\kappa$-ary site which has all finite local $\kappa$-prelimits (i.e. whenever $D$ is a finite category).

Examples

• If $C$ has a trivial topology, then a local unary prelimit (i.e. $\kappa=\{1\}$) is precisely a weak limit. The trivial topology is always weakly $\kappa$-ary, so a trivial site is unary just when it has weak limits.

• Any limit is, in particular, a local $\kappa$-prelimit. Thus, any weakly $\kappa$-ary site with finite limits is $\kappa$-ary.

• If the class of all cones over $G$ is $\kappa$-ary, then it is a local $\kappa$-prelimit. Thus, any $\kappa$-small and weakly $\kappa$-ary site is $\kappa$-ary. In particular, any small site is an infinitary site.

• The regular topology on a regular category (including an exact category) is unary.

• The coherent topology on a coherent category (including a pretopos) is finitary.

• Generalizing the previous two examples, the class of all $\kappa$-small and effective-epic families on a ∞-ary regular category (including a ∞-ary exact category) is a $\kappa$-ary topology. This is called its $\kappa$-canonical topology.

• The extensive topology on a (finitary) extensive category is finitary.

• The canonical topology on any Grothendieck topos is infinitary.

• The Zariski topology on $CRing^{op}$ is finitary.

The 2-category of $\kappa$-ary sites

The 2-category $SITE_\kappa$ has $\kappa$-ary sites as its objects, and morphisms of sites as its morphisms, where we use the more general covering-flat definition of a morphism of sites.

Properties

• $SITE_\kappa$ is equivalent to a 2-category of framed allegories?; see (Shulman).

• $SITE_\kappa$ contains, as a full reflective sub-2-category, the 2-category of ∞-ary exact categories with their $\kappa$-canonical topologies. The reflector is called exact completion. When $\kappa$ is the size of the universe, this reflector applied to a small (hence infinitary) site constructs its topos of sheaves.

References

• Michael Shulman, “Exact completions and small sheaves”. Theory and Applications of Categories, Vol. 27, 2012, No. 7, pp 97-173. Free online