Discussing the role of empty algebras across the communities of universal algebra and category theory is not always easy. The issue is tied so closely to established conventions that, at times, even making those conventions explicit can feel almost taboo.

I have recently completed a paper arising from a question raised by Paolo Aglianò: does there exist a protomodular variety without an initial object?

What appears at first to be a single question turns out to conceal several distinct ones. Their precise formulation depends on whether empty algebras are admitted, on which notion of variety or prevariety is being used, and on whether finite completeness is required. Once these assumptions are made explicit, the original question separates into eight different cases, with different answers.

The paper is written as a dialogue. This form is not merely stylistic: it allows the assumptions to emerge gradually, separates questions that are easily conflated, and makes the process of conceptual clarification part of the mathematical argument itself.

Among the examples considered, the category of non-empty heaps plays a particularly illuminating role. It provides a positive answer in one of the relevant settings and helps locate precisely the boundary between the different formulations of the problem.

More generally, the paper illustrates how mathematical progress may consist not only in answering a question, but also in discovering its correct formulation. Here, the answer becomes visible only after the apparently single problem has been disentangled into its distinct components.

The paper is available here

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