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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:vurp\{ 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

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

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

  • If the class of all cones over GG is κ\kappa-ary, then it is a local κ\kappa-prelimit. Thus, any κ\kappa-small and weakly κ\kappa-ary site is κ\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 κ\kappa-small and effective-epic families on a κ-ary regular category (including a κ-ary exact category) is a κ\kappa-ary topology. This is called its κ\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 opCRing^{op} is finitary.

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

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

  • SITE κSITE_\kappa contains, as a full reflective sub-2-category, the 2-category of κ-ary exact categories with their κ\kappa-canonical topologies. The reflector is called exact completion. When κ\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

Revised on September 6, 2012 19:47:05 by Mike Shulman (71.136.235.154)