Given an algebraic signature $\sigma$ having precisely one constant symbol, is it true that if $A$ is a set of quasi-identities in the language of $\sigma$, then the set-theoretic models of $(\sigma,A)$ will always form a pointed category? I think so, since the unique model whose underlying set has precisely one element should serve as the zero object.
2026-03-30 17:00:17.1774890017
Signatures having precisely one constant symbol, and pointed categories.
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A category with a terminal object is pointed if and only if terminal objects are initial. But in general the algebra with one element need not be freely generated by $\emptyset$. For instance, consider the language with one constant and one binary operation and no axioms; then the algebra freely generated by $\emptyset$ is the free magma generated by one element, i.e. the set of rooted binary trees.