A general Stone-Gel’fand duality

Abstract:

We give a simple characterization of full subcategories of equational categories. If $\mathcal {a}$ is one such and $\mathcal {B}$ is the category of topological spaces, we consider a pair of adjoint functors ${\mathcal {a}^{op}}\underset {F}{\overset {U}{\longleftrightarrow }}\mathcal {B}$ which are represented by objects I and J in the sense that the underlying sets of $U(A)$ and $F(B)$ are $\mathcal {a}(A,I)$ and $\mathcal {B}(B,J)$. (One may take I and J to have the same underlying set.) Such functors always establish a duality between Fix FU and Fix UF. We study conditions under which one can conclude that FU and UF are reflectors into Fix FU and Fix UF, that Fix FU = Image F = the limit closure of I in $\mathcal {a}$ and that Fix UF = Image U = the limit closure of J in $\mathcal {B}$. For example, this happens if (1) $\mathcal {a}$ is a limit closed subcategory of an equational category, (2) J is compact Hausdorff and has a basis of open sets of the form $\{ x \in J|\alpha (I)(x) \ne \beta (I)(x)\}$, where $\alpha$ and $\beta$ are unary $\mathcal {a}$-operations, and (3) there are quaternary operations $\xi$ and $\eta$ such that, for all $x \in {J^4},\xi (I)(x) = \eta (I)(x)$ if and only if ${x_1} = {x_2}$ or ${x_3} = {x_4}$. (The compactness of J may be dropped, but then one loses the conclusion that Fix FU is the limit closure of I.) We also obtain a quite different set of conditions, a crucial one being that J is compact and that every f in $\mathcal {B}({J^n},J)$, n finite, can be uniformly approximated arbitrarily closely by $\mathcal {a}$-operations on I. This generalizes the notion of functional completeness in universal algebra. The well-known dualities of Stone and Gelfand are special cases of both situations and the generalization of Stone duality by Hu is also subsumed.