I am trying to read this paper, but I have no knowledge of coalgebra and have just started to learn Category Theory so I am struggling to understand it.
Are there any references that can explain coalgebra more simply to someone who has little to no knowledge of category theory and no knowledge of universal algebra and coalgebra?
On
I would suggest: Hopf algebras: an introduction, Sorin Dascalescu, Constantin Nastasescu, Serban Raianu
It starts from the "ground" and the first chapters introduce the basic notions with minimal prerequisites (mainly linear algebra, and a first couse in abstract algebra). It emphasizes on structures and costructures over fields and its first chapters elaborate on the duality between algebras and coalgebras and the motivations for the introduction of the definitions. Contains a wealth of examples and detailed calculations. It also covers the elementary notions of Category theory in a separate and well-written appendix.
I would suggest 2 references:
"Quantum Groups", Kassel
"Algebraic Operads", Loday & Vallette
The book by Kassel is very well written and contains an expository part on algebras / coalgebras with modules / comodules as well. The second reference is an advanced book on algebraic structures called operads. Chapter I ("Algebras, coalgebras, homology") contains, in particular, an introduction to coalgebras.