The recognition principle

The recognition principle makes precise the analogy between spectra and abelian groups. In this chapter, we define an \(\infty \)-category \(\CGrp (\An )\) of commutative groups in \(\An \), also known as \(E_{\infty }\)-groups, and construct a fully faithful functor \[ \bB ^{\infty }\colon \CGrp (\An ) \hookrightarrow \Sp \] whose image is the full subcategory \(\Sp _{\geq 0}\) of connective spectra. This result is classically known as the recognition principle for infinite loop spaces.

The \(\infty \)-categorical definition of commutative groups is more subtle than the one from classical algebra. Homotopical analogues of familiar algebraic structures come with infinite hierarchies of coherence conditions, making it impossible to define them by simply writing down a list of relations. Following ideas of Segal, we take a direct approach¹ : we define (commutative) monoids and groups as functors out of suitable indexing \(\infty \)-categories, namely the opposite simplex category \(\simp \catop \) and the span category of finite sets \(\Span (\Fin )\). This functor-based perspective elegantly encodes all required coherences at once.

The equivalence between commutative groups in \(\An \) and connective spectra is a consequence of another result called the recognition principle for loop spaces. For every pointed anima \(X\) the loop space \(\Omega X\) inherits the structure of a group object in \(\An _*\). The recognition principle says that, conversely, every group object \(G\) arises this way: there exists a pointed anima \(X\) with \(G \simeq \Omega X\), and if we require \(X\) to be connected then it is unique. This \(X\) is called the delooping of \(G\) and is denoted \(\bB G\).

If \(G\) is in fact a commutative group object, the delooping \(\bB G\) is again a commutative group, so the construction can be iterated: \(\bB ^2 G := \bB (\bB G)\), and so on. The resulting sequence of deloopings assembles into a connective spectrum \(\bB ^{\infty } G\). Conversely, the underlying anima of a spectrum comes with a preferred commutative group structure, which for \(\bB ^{\infty }G\) recovers \(G\).

This chapter is organized as follows. We introduce monoids and groups in Section 5.1, and treat the recognition principle for loop spaces in Section 5.2. We then introduce commutative monoids and groups in Section 5.3, and formulate and prove the recognition principle for infinite loop spaces in Section 5.4. Finally, Section 5.5 gives an explicit telescope model for group completions under a useful hypothesis.

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