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A unified framework for notions of algebraic theory

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Soichiro Fujii

Universal algebra uniformly captures various algebraic structures, by
expressing them as equational theories or abstract clones. The ubiquity of
algebraic structures in mathematics and related fields has given rise to
several variants of universal algebra, such as theories of symmetric
operads, non-symmetric operads, generalised operads, PROPs, PROs, and
monads. These variants of universal algebra are called {notions of
algebraic theory}. In this paper, we develop a unified framework for them.
The key observation is that each notion of algebraic theory can be
identified with a monoidal category, in such a way that algebraic theories
correspond to monoid objects therein. To incorporate semantics, we
introduce a categorical structure called {metamodel}, which formalises a
definition of models of algebraic theories. We also define morphisms
between notions of algebraic theory, which are a monoidal version of
profunctors. Every strong monoidal functor gives rise to an adjoint pair
of such morphisms, and provides a uniform method to establish isomorphisms
between categories of models in different notions of algebraic theory. A
general structure-semantics adjointness result and a double categorical
universal property of categories of models are also shown.

Keywords:
Algebraic theories, clones, operads, double limits

2010 MSC:
18C10, 18D05

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 40, pp 1246-1316.

Published 2019-11-25.

http://www.tac.mta.ca/tac/volumes/34/40/34-40.pdf

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