Lawvere et al, algebraic theories (changes)

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- Lawvere et al, algebraic theories, Cambridge University Press 2010

From the two categorical approaches to algebra: algebraic theories (aka. Lawvere theories) and monads, the book -with the exception of Appendix A and Appendix C- entirely concentrates on algebraic theories. Also the book only treats sets and “many sorted sets”. Sets with structure such as topological groups are not treated.

Finitary monads for $K=Set$ yield precisely one-sorted algebraic theories.

Finitary monads for $K=Set^S$ yield precisely $S$-sorted algebraic theories.

(This (or parts thereof) was first shown in: Linton, F. E. J., Some aspects of equational theories, Proc. Conf. on Categorical Algebra at La Jolla (1966), 84–95. For the history see Martin Hyland, The category theoretic understanding of universal algebra Lawvere theories and monads.)

Proposition: For every one-sorted algebraic theory $(T, t)$, the concrete category $(Alg T, Alg t)$ is pseudomonadic.

Theorem: The following conditions on a concrete category $(A,U)$ over $Set$ are equivalent:

(1) $(A,U)$ is pseudo-one-sorted algebraic.

(2) $A$ is cocomplete, and $U$ is a conservative right adjoint preserving sifted colimits.

Corollary: Pseudo-one-sorted algebraic categories are up to pseudoconcrete equivalence precisely the categories $Set^M$ of Eilenberg-Moore algebras for finitary monads $M$ on $C$.

Theorem (One-sorted algebraic duality): The category $Alg_u^1$ of uniquely transportable on-sorted algebraic categories is equivalent to the dual of the category $Th^1$ of one-sorted algebraic theories. In fact, the $2$-functor $Alg^1:(Ps Th^1)^{op}\to Ps Alg^1$ is a biequivalence.

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