Bases as coalgebras (Q5891175)

Let \(T\) be a monad on a category \(\mathbf K\), then \(T\) induces a comonad \(\bar {T}\) on the category of algebras of the monad \(T\). The comonad \(\bar {T}\) induces a monad \(\bar{\bar {T}}\) on the category of coalgebras of \( \bar {T}\) and so on. It is shown that coalgebras over \(\bar {T}\) model bases. Algebras over \(T\) are used for composition and coalgebras for decomposition. A monad \((T,\eta ,\mu )\) on a poset-enriched category \( \mathbf A\) is a Kock-Zöberlein monad if \(T:\mathbf A(X,Y)\rightarrow\mathbf A(TX,TY)\) is monotone and \(T\eta_X\leq\eta_{TX}\). If \(T\) is a Kock-Zöberlein monad then for a morphism \(a:TX\rightarrow X\) in \(\mathbf A\) the following are equivalent: a) \(a:TX\rightarrow X\) is an Eilenberg-Moore algebra of \(T\); b) \(a\) is a left-adjoint-left-inverse of the unit \(\eta :X\rightarrow TX\). If \(a:TX\rightarrow X\) is an algebra of \(T\) then for a morphism \(c:X\rightarrow TX\) the following are equivalent: a) \(c:a\rightarrow\bar {T}a\) is an Eilenberg-Moore coalgebra of \(\bar {T}\); b) \(c\) is a left-adjoint-right-inverse of the counit \(a:\bar {T}a\rightarrow a\). Further \(T:\mathrm{Alg}(T)\rightarrow\mathrm{Alg}(\bar{\bar { T}})\) is an equivalence of categories and thus the construction of monads and comonads stops after the second step. These facts are applied to vector spaces, complete lattices, convex sets, dcpos over posets, frames over semilattices, and the sematics of programs in the Kleisli category of a monad. These notions are used for a description of the comonoid structure describing orthonormal bases in finite-dimensional Hilbert spaces.