Multiplicative structures on Thom spectra

Construction 10.2.8 (Multiplicative structures on Thom spectra). Let \(A\) be a small commutative monoid in animae, regarded as a symmetric monoidal \(\infty \)-category, and let \[ \xi \colon A\longrightarrow \Pic (\S )\subseteq \Sp \] be a symmetric monoidal functor. We equip the Thom spectrum \(\th (\xi )\) with a canonical commutative ring spectrum structure, following Antolín-Camarena and Barthel (2019).

By Corollary 16.2.8, the functor category \(\Fun (A,\Sp )\) admits the Day convolution symmetric monoidal structure. The universal property of Day convolution from Lemma 16.1.4 identifies the underlying lax symmetric monoidal functor of \(\xi \) with a commutative algebra in \(\Fun (A,\Sp )\). The colimit functor \[ \colim _A\colon \Fun (A,\Sp )\longrightarrow \Sp \] is pointwise left Kan extension along the unique symmetric monoidal functor \(A\to *\), so it is symmetric monoidal by Lemma 16.2.11. Here \(\Fun (*,\Sp )\) is identified with \(\Sp \) equipped with its usual tensor product. It therefore carries \(\xi \) to a commutative algebra in spectra whose underlying spectrum is \[ \colim _A\xi =\th (\xi ). \]