Higher theories of algebraic structures

Abstract

The notion of (symmetric) coloured operad or "multicategory" can be obtained from the notion of commutative algebra through a certain general process which we call "theorization" (where our term comes from an analogy with William Lawvere's notion of algebraic theory). By exploiting the inductivity in the structure of higher associativity, we obtain the notion of "$n$-theory" for every integer $n\ge 0$, which inductively "theorizes" $n$ times, the notion of commutative algebra. As a result, (coloured) morphism between $n$-theories is a "graded" and "enriched" generalization of ($n-1$)-theory. The inductive hierarchy of those "higher theories" extends in particular, the hierarchy of higher categories. Indeed, theorization turns out to produce more general kinds of structure than the process of categorification in the sense of Louis Crane does. In a part of low "theoretic" order of this hierarchy, graded and enriched $1$- and $0$-theories vastly generalize symmetric, braided, and many other kinds of enriched multicategories and their algebras in various places. We make various constructions of/with higher theories, and obtain some fundamental notions and facts. We also find iterated theorizations of more general kinds of algebraic structure including (coloured) properad of Bruno Vallette and various kinds of topological field theory (TFT). We show that a "TFT" in the extended context can reflect a datum of a very different type from a TFT in the conventional sense, despite close formal similarity of the notions. This work is intended to illustrate use of simple understanding of higher coherence for associativity.