Truncation and connectivity

In Chapter 2, we saw that the effective epimorphisms and monomorphisms form a factorization system on every topos. This is the first stage of a hierarchy: effective epimorphisms are precisely the \((-1)\)-connected maps, while monomorphisms are precisely the \((-1)\)-truncated maps. In this chapter, we develop the corresponding notions of \(n\)-connected and \(n\)-truncated morphisms for every \(n \geq -2\).

Truncation is defined recursively by diagonals: a morphism is \(n\)-truncated if its diagonal is \((n-1)\)-truncated. In a topos, the \(n\)-truncated objects form a reflective subcategory, and truncation commutes with base change. Connectivity is then characterized by orthogonality to truncated maps. The main structural result of Section 3.1 and Section 3.2 is that the \(n\)-connected and \(n\)-truncated morphisms form a factorization system for every \(n\). These compatible factorizations provide the internal Postnikov calculus of a topos.

In Section 3.3, we attach homotopy group objects \(\pi_n(X) \in T_{/X}\) to every object \(X \in T\). They fit into long exact sequences and detect truncation and connectivity. In particular, a morphism is \(n\)-connected if and only if it is an effective epimorphism and its diagonal is \((n-1)\)-connected. This recursive criterion is the connected analogue of the recursive definition of truncated maps.

Passing from finite \(n\) to \(\infty\) leads to hypercompletion. A morphism is \(\infty\)-connected if it is \(n\)-connected for every \(n\), and hypercompletion is the left exact localization which forces all such morphisms to become isomorphisms. Thus hypercompleteness asks whether a morphism which is invisible to every finite stage of the truncation hierarchy is already an isomorphism. We develop this in Section 3.4.

We conclude in Section 3.5 with objects lying simultaneously at adjacent levels of the two hierarchies. An \(n\)-gerbe is \(n\)-truncated and \((n-1)\)-connected; a pointed \(n\)-gerbe is an Eilenberg–MacLane object. Iterated delooping identifies these objects with iterated group objects. For \(n \geq 1\) and a static abelian group object \(A\), the universal banded \(n\)-gerbe

classifies \(n\)-gerbes banded by an abelian group object \(A\). This gives an intrinsic construction of cohomology with coefficients in \(A\) inside an arbitrary topos.