symmetric monoidal (∞,1)-category of spectra
A convolution product is the binary operation on ring-valued (or more generally magma-valued) functions $f$ on a group $G$ or more generally on the set of morphisms $\mathcal{G}$ of a groupoid, which is given by summing or more generally integrating products of values on complementary elements of the schematic form
where $h^{-1}$ denotes the inverse element or more generally the inverse morphism of $h$.
For example if $G$ is a finite group, then this defines the group algebra, or more generally if $G$ is a suitable topological group or Lie group or more generally a suitably topological groupoid or Lie groupoid, then an integral-version of this formula defines the groupoid convolution algebra.
Often the convolution product is considered in analysis for suitably integrable functions $f \colon \mathbb{R} \to \mathbb{R}$ on the real line $\mathbb{R}^1$ regarded as a group with respect to addition of real numbers:
This generalizes to a convolution product of distributions. Thes convolution products plays a central role in Fourier analysis
A categorification of the concept of convolution is the concept of Day convolution.