symmetric monoidal (∞,1)-category of spectra
There are many related constructions of algebras, topological algebras and so on which bear the name of a convolution algebra.
The basic mechanism is usually the
The probably most widespread example of this is the
This is a special case of the
Let $k$ be a commutative unital ring, $(C,\Delta)$ a (counital) $k$-coalgebra and $(A,m)$ an associative (unital) $k$-algebra. Then the set of linear maps
has a structure of an associative (unital) algebra, called convolution algebra, in which the product of two linear maps $f,g$ is given by
Let $(\mathcal{C}, \otimes, I, \multimap)$ be a closed monoidal category, $(A,\Delta,\eta)$ a comonoid in $\mathcal{C}$ and $(B,\nabla,\epsilon)$ a monoid. Then $A \multimap B$ is a monoid.
Let $(\mathcal{C}, \otimes, I, \multimap)$ be a closed monoidal category, $Mon(\mathcal{C})$ the category of monoids of $\mathcal{C}$ and $Comon(\mathcal{C})$ the category of comonoids? of $\mathcal{C}$. We then have a functor:
which associate $A \multimap B$ to $(A,B)$
Given a finite group $G$ and a ring $R$, the space of functions $C(G,R)$ inherits the convolution product defined by
This is the non-commutative product operation that appears in the Hopf algebra structure on $C(G,R)$.
More generally, there is convolution of functions on morphisms of a groupoid. See at groupoid convolution algebra for details.
Last revised on November 25, 2022 at 18:04:35. See the history of this page for a list of all contributions to it.