Higher Lawvere theories

Abstract

We survey Lawvere theories at the level of 8-categories, as an alternative framework for higher algebra (rather than 8-operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also better-suited than operads for equivariant homotopy theory and its relatives.Our main result establishes a universal property for the 8-category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its 8-category of models. Many familiar properties of Lawvere theories follow directly.As a consequence, we prove that the Burnside category is a classifying object for additive categories, as promised in an earlier paper, and as part of a more general correspondence between enriched Lawvere theories and module Lawvere theories.

On the commutative algebra of categories

“…However, we will delay the last result (that Burn-modules are additive categories) until a sequel [5]. Although it is possible to prove using similar techniques to those in this paper (and we invite the reader to do so), the proof is more ad hoc.…”

“…Moreover, for each semiring category R on this list, the forgetful functor Mod R Ñ SymMon is fully faithful; that is, being an R-module is a property of a symmetric monoidal category, rather than containing extra structure. These properties are as follows: However, we will delay the last result (that Burn-modules are additive categories) until a sequel [5]. Although it is possible to prove using similar techniques to those in this paper (and we invite the reader to do so), the proof is more ad hoc.…”

“…Various notions of categorified rings have appeared before, including to study iterated K-theory [1], Tannaka duality [18], and smash products of spectra ( [22] 4.8). The framework in this paper is general enough to permit comparisons to many of the other notions of categorified rings; for the most part, we will not discuss this, but the comparison to what may be called presentable categorified rings (as in [22] 4.8 and [18]) is touched on in Remark 3.5 and Lemma 3.7, and the author hopes to elaborate on this relationship in a sequel [5].…”

“…In 3.3 we discuss the other possible properties from the list above -with the exception of additive 8-categories themselves, which will be addressed in a sequel work [5].…”

“…This construction is known to many people, but we are not aware of a reference for 8-categories. It will appear in detail in the sequel to this paper [5].…”