Lawvere-Tierney sheaves in algebraic set theory

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

💡 Research Summary

The paper addresses the problem of defining internal sheaves within Algebraic Set Theory (AST) in a way that parallels the well‑established construction in elementary topos theory. In the topos setting, internal sheaves are obtained from a Lawvere‑Tierney (LT) local operator on the subobject classifier; the authors replace this operator by a more general notion of a Lawvere‑Tierney coverage, which is a subobject of the object of small truth values Ω_S.

The authors begin by fixing a Heyting pretopos E equipped with a family S of “small maps”. The small‑map axioms (A1)–(A7) are the usual closure, pull‑back stability, descent, unit, coproduct, quotient, and collection conditions introduced by Joyal‑Moerdijk. In addition they assume the power‑object axiom (P1), which guarantees for each object X a power object P(X) together with a universal membership relation, allowing every indexed family of small subobjects to be classified. This axiom is equivalent to the existence of exponentials for small maps (S1) together with a weak representability condition (S2); the latter only requires a quasi‑pullback rather than a genuine pullback, which is essential for working in exact completions without extra strength.

With these preliminaries, the object of small truth values is defined as Ω_S = P(1). A Lawvere‑Tierney coverage J ⊆ Ω_S is a subobject satisfying two internal logical conditions: (C1) J holds of the top element ⊤, and (C2) for all p,q∈Ω_S, (p⇒J(q)) ⇒ (J(p)⇒J(q)). When E is an elementary topos and S consists of all morphisms, this definition coincides with the usual LT local operators via the universal property of Ω_S as the subobject classifier.

From a coverage J the authors construct a closure operator on subobjects of any X∈E, using the collection axiom to form the “J‑closure” of a subobject. An object X equipped with a J‑closed subobject structure is called a J‑sheaf. The main technical achievement is showing that the full subcategory Sh_J(E) of J‑sheaves is again a Heyting pretopos and that the inclusion i:Sh_J(E)↪E admits a finite‑limit‑preserving left adjoint a_J (the associated sheaf functor). The construction of a_J proceeds exactly as in the topos case: first form the J‑closure of the diagonal, then take the coequaliser of the resulting equivalence relation. The crucial point is that all these steps stay within the realm of small maps, thanks to the collection axiom and the power‑object axiom.

Moreover, the family S of small maps lifts to Sh_J(E): the authors prove that locally small maps between sheaves satisfy the same axioms (A1)–(A7) and (P1) inside the sheaf category. Consequently, the pair (Sh_J(E),S_J) is again an AST‑model, preserving the logical strength of the original one.

Two ambient settings are discussed. The “exact setting” assumes only that E is a Heyting pretopos and S satisfies (A1)–(A7) plus (P1). The “bounded exact setting” adds that every diagonal Δ_X is small, that small maps are stable under quotients, and that E has quotients of bounded equivalence relations. The paper works in the exact setting because it is more general and accommodates examples where the diagonal need not be small (e.g., certain constructive set theories).

The authors illustrate the theory with three examples. (1) In an elementary topos with all morphisms small, the construction recovers the classical LT sheaves. (2) For Constructive Zermelo‑Fraenkel set theory (CZF), they take E to be the exact completion of the category of classes; small maps are those whose fibres are sets. This example satisfies (A1)–(A7) and (S1)–(S2) but not the stronger representability axiom, showing the necessity of the weakened condition. (3) For Martin‑Löf type theory, they use the category of setoids; small maps are those that factor through a quasi‑pullback with a fibre isomorphic to a type‑universe. Again the weak representability suffices.

In summary, the paper provides a robust, axiomatic framework for internal sheaves in Algebraic Set Theory based on Lawvere‑Tierney coverages. By weakening the small‑map axioms to the essential (A1)–(A7) plus (P1) and using the collection axiom, the authors obtain a sheaf construction that works in a wide range of constructive and categorical models, including those arising from CZF and type‑theoretic setoids. The results generalize the classical topos‑theoretic sheaf theory, preserve the small‑map structure, and open the way for further applications such as realizability models, coalgebraic constructions, and W‑type analysis within AST.