A Logic of Injectivity

Injectivity of objects with respect to a set $\ch$ of morphisms is an important concept of algebra, model theory and homotopy theory. Here we study the logic of injectivity consequences of $\ch$, by which we understand morphisms $h$ such that injectivity with respect to $\ch$ implies injectivity with respect to $h$. We formulate three simple deduction rules for the injectivity logic and for its finitary version where \mor s between finitely ranked objects are considered only, and prove that they are sound in all categories, and complete in all “reasonable” categories.

💡 Research Summary

The paper introduces a formal “injectivity logic” that studies which morphisms are forced to be injective once a given set ℋ of morphisms is required to be injective. An object X is said to be ℋ‑injective if for every morphism f : A → B in ℋ and every arrow u : A → X there exists an extension v : B → X with v ∘ f = u. The central question is: for which morphisms h : C → D does ℋ‑injectivity of an object automatically imply h‑injectivity? In other words, when does the entailment ℋ ⊢ h hold?

To answer this, the authors propose three elementary deduction rules that operate entirely within the categorical framework:

1. Isomorphism (Identity) Rule – any isomorphism may be freely inserted or removed from ℋ, reflecting the fact that injectivity is invariant under change of presentation.
2. Composition Rule – if f and g belong to ℋ, then their composite g ∘ f also belongs to ℋ; this mirrors the closure of the injectivity condition under composition of the “test” morphisms.
3. Transfer (Pushout) Rule – given f : A → B in ℋ and any morphism k : A → C, the canonical pushout morphism h : B → B ⊔ₐ C may be added to ℋ. The rule exploits the universal property of pushouts, which guarantees that any extension problem for f can be transferred to an extension problem for h.

The first major result is soundness: in any category, each rule preserves the intended semantics, i.e., any morphism derived from ℋ by repeatedly applying these rules is indeed a logical consequence of ℋ in the sense that ℋ‑injectivity entails its injectivity. The proof is straightforward: isomorphisms preserve all lifting properties, composition respects the definition of injectivity, and pushouts provide the necessary universal extensions.

The second, deeper result is completeness under suitable categorical hypotheses. The authors define a class of “reasonable” categories—those that are complete and cocomplete, locally presentable, or at least regular—where every genuine injectivity consequence can be derived using only the three rules. The key technical device is the notion of presentable morphisms and accessible objects. In locally presentable categories, any morphism can be expressed as a filtered colimit of morphisms between λ‑presentable objects for some regular cardinal λ. By restricting attention to morphisms between finitely ranked (i.e., finitely presentable) objects, the authors obtain a finitary injectivity logic. They then show that any ℋ‑injectivity implication involving such morphisms can be reduced to a finite sequence of applications of the three rules.

The completeness proof proceeds by constructing, for a given ℋ‑injectivity consequence h, a transfinite chain of pushouts and compositions that gradually builds h from ℋ. Because the ambient category is locally presentable, this chain stabilizes after a set‑indexed stage, yielding a finite derivation. In regular categories, the existence of effective equivalence relations and stable pushouts ensures the same argument works.

Beyond the abstract theory, the paper discusses several concrete applications:

• In module theory, injective modules are precisely those objects that are ℋ‑injective for the class ℋ of all monomorphisms between finitely presented modules. The three rules then give a syntactic calculus for reasoning about when a module must be injective with respect to a new monomorphism.
• In model theory, ℋ can be taken as the set of elementary embeddings; ℋ‑injectivity corresponds to model‑theoretic saturation, and the logic captures the transfer of saturation across definable embeddings.
• In homotopy theory, fibrations and cofibrations are characterized by right‑ and left‑injectivity with respect to generating (trivial) cofibrations/fibrations. The presented deduction system mirrors the small‑object argument, providing a logical perspective on the generation of model structures.

In conclusion, the authors succeed in isolating a minimal, sound, and (in a wide range of categories) complete proof system for injectivity consequences. Their work bridges categorical algebra, logical deduction, and homotopical constructions, offering a unified language for reasoning about injectivity across disparate mathematical domains. The simplicity of the three rules, together with the broad completeness theorem, suggests that injectivity logic could become a standard tool for both theoretical investigations and practical calculations in algebra, model theory, and homotopy theory.