Pretopoi isolate the finitary operations which preserve coherent objects: fiber products, finite coproducts, and effective quotients of groupoid objects. Thus the coherent objects of a coherent topos form its finitary or syntactic core, while sheafification for the effective epimorphism topology reconstructs the ambient topos. The main result of this section makes this relationship precise in the bounded and hypercomplete settings. It also prepares for Makkai completeness: points are models of the pretopos of coherent objects, and ultraproducts retain enough structure to recover that syntax.
In a local pretopos, we have the factorization system (effective epi, mono). Also the factorization system (\(n\)-connected, \(n\)-truncated) exists, but this is less obvious. It amounts to showing that \(n\)-truncation exists.
Let \(C\) be a local pretopos. The effective epi topology on \(C\) is defined as follows: a sieve on \(X \in C\) is declared to be a covering sieve if it contains a finite family \((Y_i \to X)_i\) such that the map \(\bigsqcup_i Y_i \to X\) is an effective epimorphism.
If \(C\) is small, we obtain a topos \(\Shv(C)\) with respect to this topology.
If \(T\) is a topos, then \(T^{\coh}\) is a local pretopos. In particular, if \(T\) is coherent, then \(T^{\coh}\) is a pretopos.
Proof
Since \(T\) satisfies conditions (1)â(3), it suffices to show that \(T^{\coh}\) is closed under the following operations:
Fiber products: This is clear from the definition of coherent objects.
Finite coproducts: This follows by induction, reducing in degree zero to the stability of quasi-compactness under finite coproducts.
Colimits of groupoids: Let \(X_{\bullet}\) be a groupoid object with realization \(X\). The square is a pullback and \(X_0\to X\) is an effective epimorphism. The descent criterion for relative coherence, applied successively in every degree, therefore shows that \(X\) is coherent whenever \(X_0\) and \(X_1\) are coherent. This is [Lurie 2018, Proposition A.6.1.6].
A prelogos is just a pretopos. A morphism of prelogoi from \(S\) to \(T\) is a functor \(f^*\colon S \to T\) which preserves finite limits, finite coproducts, and colimits of groupoids. This defines a category \(\Logos^{\pre}\) of prelogoi. The category of pretopoi is defined as \(\Topos^{\pre} := (\Logos^{\pre})\catop\).
We denote by \(\Topos^{\coh} \subseteq \Topos\) the non-full subcategory spanned by the coherent topoi and the coherent morphisms: those morphisms of topoi \(f\colon T \to S\) such that \(f^*\colon S \to T\) preserves coherent objects.
There are functors
\[\Topos^{\coh} \to \Topos^{\pre}, \qquad T \mapsto T^{\coh}\]
and
\[\Topos^{\pre} \to \Topos^{\coh}, \qquad C \mapsto \Shv(C).\]
A pretopos \(C\) is called bounded if it is small and every \(X \in C\) is truncated.
A pretopos \(C\) is hypercomplete if the following conditions are satisfied:
It admits colimits of Kan simplicial objects, meaning simplicial objects whose matching maps satisfy the internal Kan lifting condition.
If \(X_{\bullet} \to X\) is a trivial Kan fibration, meaning a Kan fibration whose relative homotopy objects are all trivial, then \(\abs{X_{\bullet}} \simeq X\).
\[\Topos^{\coh,b} \simeq \Topos^{\pre,b}, \qquad T \mapsto T^{\coh}_{< \infty}, \qquad C \mapsto \Shv(C)\]
and
\[\Topos^{\coh,\text{loc coh}, \hyp} \simeq \Topos^{\pre,\hyp}, \qquad T \mapsto T^{\coh}, \qquad C \mapsto \Shv(C)^{\hyp}.\]
These equivalences are induced by the effective epimorphism topology and are proved in [Lurie 2018, Sections A.6.5--A.7.4]. In the second equivalence, local coherence ensures that \(T^{\coh}\) is essentially small; hypercompleteness is exactly the additional descent condition needed to recover \(T\) from this pretopos.
References
Jacob Lurie. Spectral Algebraic Geometry. under construction (version dated February 2018), www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf. 2018.
