Lawvere et al, algebraic theories

This entry is about

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.

Appendix A

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

Finitary monads for K=Set SK=Set^S yield precisely SS-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.)

Appendix C

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

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

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

(2) AA is cocomplete, and UU is a conservative right adjoint preserving sifted colimits.

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

Theorem (One-sorted algebraic duality): The category Alg u 1Alg_u^1 of uniquely transportable on-sorted algebraic categories is equivalent to the dual of the category Th 1Th^1 of one-sorted algebraic theories. In fact, the 22-functor Alg 1:(PsTh 1) opPsAlg 1Alg^1:(Ps Th^1)^{op}\to Ps Alg^1 is a biequivalence.