The other pullback lemma

Abstract: Given a composition of two commutative squares, one well-known pullback lemma says that if both squares are pullbacks, then their composition is also a pullback; another well-known pullback lemma says that if the composed square is a pullback and the supporting square is a pullback, then the other square is a pullback as well. This note investigates the third of the possibilities.

• The paper establishes necessary and sufficient conditions for the transfer of pullback properties using strong, extremal, and regular epimorphisms.
• It presents counterexamples in the categories of Graph and Pos to illustrate failures of the lemma under common morphism assumptions.
• It reformulates the lemma in a fibrational framework, linking pullback properties to the conservativity of reindexing functors and adjunctions.

Summary of "The Other Pullback Lemma" (1311.2974)

This work investigates structural properties of pullbacks in categorical contexts, focusing on conditions under which the so-called "other pullback lemma" holds. While the composition of pullback squares and inference of the left inner square from the outer square and right square being pullbacks are classical results, this note scrutinizes the converse scenario. Specifically, it formally analyzes when, given a commutative composition of two squares, the pullback property of the left and the outer squares implies the pullback property of the right square. The analysis is furnished with counterexamples, a comprehensive categorical characterization, and reformulation in the geometric language of fibrations.

Analysis of Pullback Lemmata and Counterexamples

The classic "pullback pasting lemmas" can be summarized as follows: if (I) and (II) are pullbacks, then the outer square is a pullback; and if the outer and right square (II) are pullbacks, so is the left square (I). The paper addresses the converse: does the pullback property of the left inner and the outer square entail that the right square is a pullback? The answer is shown to be negative in general. Explicit counterexamples are constructed in the categories of simple graphs and posets (Graph and Pos), indicating the failure of the "other pullback lemma" even with strong assumptions on morphisms (e.g., epimorphisms, regular epis).

The first counterexample in Graph demonstrates the failure with an injective-on-nodes homomorphism, showing that even when the left morphism is an epi, the right square need not be a pullback. The second counterexample in Pos uses coequalisers to construct a scenario where a regular epimorphism does not satisfy the "other pullback lemma." This is further used to show that regularity is not ensured in Pos, contrasting with regular categories.

Categorical Characterization and Main Theorem

Fundamental definitions of extremal, strong, and regular epimorphisms are recalled. The equivalence of the following statements is established for a morphism $e: X \to Y$ in a category with pullbacks:

• The other pullback lemma holds along $e$
• $e$ is a strong morphism stable under pullbacks
• $e$ is an extremal morphism stable under pullbacks

This is codified in Theorem 1, which provides necessary and sufficient conditions on a morphism for the validity of the "other pullback lemma." The result leverages the stability of extremal and strong morphisms under pullback and utilises well-known structural facts about regular and balanced categories.

Corollaries show that in regular categories, the property reduces to $e$ being a regular epimorphism, and in balanced regular categories (such as toposes and pretoposes, but not quasitoposes), it reduces to $e$ being an epimorphism. This aligns with the categorical folklore and clarifies the minimal hypotheses needed.

Reformulation Using Fibrational/Geometric Language

The paper transitions from elementwise (diagrammatic) reasoning to a geometric/fibrational perspective. Pullback properties are encoded in terms of cartesian morphisms for the fundamental indexing functor of the arrow category, denoted $\text{cod}(B) : B^+ \rightarrow B$.

A connection is developed between cartesian morphisms (pullbacks), conservatively cartesian morphisms, and reindexing functors. It is shown that the "other pullback lemma" can be reformulated as a conservation property of reindexing functors: a morphism $e : X \rightarrow Y$ satisfies the lemma if and only if the associated reindexing functor (from the slice over $Y$ to the slice over $X$) is conservative (reflects isomorphisms).

The theory is further strengthened via the use of adjunctions, relating conservative right adjoints to the extremality of counits in adjunctions and employing Isbell’s correction to naive adjunction folklore. This yields equivalence between conservativity of the reindexing functor and extremality of certain canonical morphisms, solidifying the main categorical result.

Implications and Theoretical Significance

The work completes the classification of pullback pasting results, filling a notable theoretical gap in the literature. These conditions have practical implications for reasoning in categorical logic, especially in regular and balanced regular categories where regular and strong epimorphisms abound (for example, toposes and algebraic categories).

From a foundational perspective, this clarifies when fibered or reindexed structures in model theory, topos theory, and categorical semantics behave well under composition and descent—a key aspect in topos-theoretic logic and regular category semantics.

In the broader context of categorical algebra and categorical logic, the fibrational reformulation streamlines reasoning about descent, glueing, and exactness properties and provides a methodology for analyzing when other structural results relying on pullback squares can be generalized or may fail.

Conclusion

This note rigorously delineates the precise categorical circumstances under which the "other pullback lemma" holds. The core result reduces the property to stability of (strong/extremal/regular) epimorphisms under pullback, and in regular or balanced categories, to regular or plain epimorphisms, respectively. The fibrational perspective not only generalizes elementwise arguments but also connects the result to properties of reindexing functors. The findings inform applications in category theory, topos theory, and categorical logic, particularly in contexts where the manipulation of pullback squares is central.