Understanding homomorphism from coalgebra to algebra

Question

Could someone please explain what exactly a homomorphism from coalgebra to algebra (from this paper: 1, page 10, definition 5.1). I understand a homomorphism as a map between two structures which preserves operations and their neutral elements, but which operations would it preserve between coalgebra and algebra? Thank you.

Answer 1

Maybe an answer is not needed anymore, because the OP question has already been solved, but I think a clarification is needed for those who are not familiar with the subject.

If we have two vector spaces $V,W$ over a field $\Bbbk$ then we can consider the set $\mathsf{Hom}_{\Bbbk}(V,W)$ of all $\Bbbk$-linear maps from $V$ to $W$.

In the particular case in which $V$ has additionally a $\Bbbk$-coalgebra structure $(V,\Delta,\varepsilon)$ and $W$ has an algebra structure $(W,m,u)$, then $\mathsf{Hom}_{\Bbbk}(V,W)$ can be endowed with a product $*$ (called the convolution product) and a unit element $1$ that makes of it a monoid, namely $$f*g:=m\circ (f\otimes g)\circ \Delta \qquad \text{and} \qquad 1:=u\circ\varepsilon,$$ which are still $\Bbbk$-linear maps from $V$ to $W$.

If $V=W=(B,m,u,\Delta,\varepsilon)$ bialgebra, then $\mathsf{Hom}_{\Bbbk}(B,B)=\mathsf{End}_{\Bbbk}(B)$ is a monoid as above and an inverse of the identity $\mathsf{Id}_B$ with respect to the convolution product (when it exists) is called and antipode. A bialgebra with an antipode is nowadays called a Hopf algebra.

Answer 2

To add to Ender's answer about the convolution algebra, let me mention that among all linear maps between (differential graded) coalgebra and algebra, there are twisting morphisms, the ones who satisfy the Maurer-Cartan equation $$ \partial(f) + f * f = 0 \ , $$ which are the heart of homotopical algebra and deformation theory, see for instance the Chapter 2 of Loday--Vallette Algebraic Operads or Dotsenko--Shadrin--Vallette Twisting procedure (to appear soon!).