Operads: the heuristics

This chapter develops the intuition behind the definition of \(\infty \)-operads before we turn to the abstract theory. Roughly speaking, an \(\infty \)-operad encodes a type of ‘algebraic structure’; for example, there will be an \(\infty \)-operad for commutative algebras, one for associative algebras, one for left modules over an associative algebra, and so on. We start with the 1-categorical situation: In Section 12.1 we introduce non-colored operads as structures encoding collections of ‘\(n\)-ary operations’ \(A^{\otimes n} \to A\), and in Section 12.2 we discuss the colored variant that allows for more general operations of the form \(A_1 \otimes \dots \otimes A_n \to B\). In order to motivate the \(\infty \)-categorical analogue of operads, we explain in Section 12.3 how to rephrase the classical notion of colored operads in a way that robustly encodes all the coherences without needing to specify them by hand. This paves the way for our definition of \(\infty \)-operads in Section 12.4. We conclude with brief pointers to other models for \(\infty \)-operads.

Generated from the authoritative LaTeX source.