Modalities

The factorization systems constructed in Chapter 3 are not isolated constructions. For every \(-2 \leq n \leq \infty\), the factorization into an \(n\)-connected map followed by an \(n\)-truncated map is preserved by base change and therefore behaves uniformly in every slice of the topos. This raises a general question: which factorization systems on a topos have this fiberwise stability, and what additional structure follows from it?

A modality is a factorization system \((L,R)\) whose left class is stable under base change. Modalities were developed systematically in a series of articles by Anel et al. (2018, 2020, 2022, 2024, 2025). They provide a common language for factorization, descent, localization, and excision phenomena in topoi.

We begin with a basic but less familiar example of a modality. In Section 5.1, we show that the epimorphisms form a modality and identify the epimorphisms in \(\An\) with the acyclic maps; the associated reflection recovers Quillen's plus construction. We then turn to excision. In Section 5.2, we prove a generalized Blakers–Massey theorem for an arbitrary modality. It gives a condition on the pushout product of the diagonals of two maps which ensures that the gap map of their pushout square lies in the left class of the modality. Its proof rests on a modal form of descent.

The next part of the chapter connects fiberwise closure properties to localizations of topoi. An acyclic class is a saturated class stable under base change, while a congruence is an acyclic class satisfying 2-out-of-3. Small generation upgrades the former to a modality and identifies the latter with the class inverted by a left exact localization. The decisive relation, proved in Section 5.3, is that the congruence generated by a set \(\Sigma\) is the acyclic class generated by all iterated diagonals of maps in \(\Sigma\):

Equivalently, congruences are precisely the acyclic classes closed under diagonals. This formula gives explicit control over left exact localization and will later yield a short proof that sheafification on a Grothendieck site is left exact.

These ideas also lead to structure theory. We construct closed complements of open immersions in Section 5.4, and in Section 5.5 we factor every morphism of logoi into a monogenic quotient, an epigenic quotient, and a conservative morphism. Finally, products of acyclic classes restrict to products of congruences and give rise to adic filtrations of topoi. Applied to the congruence of \(\infty\)-connected maps, this construction produces the Goodwillie tower of a topos; for functor topoi it recovers Goodwillie's polynomial approximations and their stable layers. This is developed in Section 5.6 and Section 5.7.

Thus the left class of every modality is an acyclic class. Conversely, an acyclic class of small generation is the left class of a modality by Proposition A.11. The distinction is useful: closure arguments can often be carried out for arbitrary acyclic classes, whereas orthogonal factorizations require a modality.