Varieties of ordered algebras as categories
Ji\v{r}\'i Ad\'amek, Ji\v{r}\'i Rosick\'y
TL;DR
This paper characterizes categories of ordered algebras as categories enriched over posets, paralleling Lawvere's classical results, and shows they are free completions of duals of Lawvere theories under sifted colimits.
Contribution
It provides a new characterization of varieties of ordered algebras as enriched categories and relates them to free completions of Lawvere theories, extending classical algebraic theory.
Findings
Categories enriched over posets are equivalent to varieties of ordered algebras.
Varieties of ordered algebras are the free completions of duals of Lawvere theories.
The characterization parallels Lawvere's classical results for ordinary algebras.
Abstract
A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere's classical characterization of varieties of ordinary algebras. We also prove that varieties of ordered algebras are precisely the free completions of duals of discrete Lawvere theories under sifted colimits.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Logic · Homotopy and Cohomology in Algebraic Topology