nLab “essentially algebraic theory”
nLab on algebraic theory
http://mathoverflow.net/questions/3003/in-what-sense-are-fields-an-algebraic-theory
See also Barr-Beck thm or something like that
Schwede has some notions for triangulated cats I think
Boerceaux vol 2 page 158: A cat equipped with a functor U to sets is called algebraic if (a) it has coequalizers and kernel pairs (b) U has a left adjoint F (c) U reflects isomorphisms (d) U preserves regular epimorphisms (e) UF preserves filtered colimits.
arXiv:1109.1598 Algebraic theories, span diagrams and commutative monoids in homotopy theory from arXiv Front: math.CT by James Cranch We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length
We study one extended example in detail: the theory of commutative monoids (which turns out to be essentially just a 2-category). This gives a straightforward, combinatorially explicit, and instructive notion of a commutative monoid. We prove that this definition is equivalent (in appropriate senses) both to the classical concept of an E-infinity monoid and to Lurie’s concept of a commutative algebra object.
nLab page on Algebraic category