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κ-ary site

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Topos Theory

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Category Theory

κ\kappa-ary sites

Idea

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

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

Definition

Let κ\kappa be an arity class.

Definition

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

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

Definition

Let CC be a site and G:DCG:D\to C a functor. A local κ\kappa-prelimit of GG is a κ\kappa-small family of cones {q i:ΔLG} i\{q_i: \Delta L \to G \}_i in CC such that for any cone r:ΔuGr:\Delta u \to G, the sieve {p:vu|rp\{ p: v\to u | r p factors through some q i}q_i \} is a covering sieve of uu.

Definition

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

Examples

The 2-category of κ\kappa-ary sites

The 2-category SITE κSITE_\kappa has κ\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

References