Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $\ast$-algebra maps over $\mathbb C$ as morphisms). The latter category can be defined in $L_{\omega_1,\omega}$ and is easily seen to be locally presentable, and in particular it is complete and cocomplete.
Recall that a topological space $X$ is called $\kappa$-Lindelöf (or $\kappa$-compact) if every open cover of $X$ admits a subcover of cardinality $<\kappa$. So compact = $\aleph_0$-Lindelöf.
Question 1: Let $\kappa$ be a cardinal. Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces complete and cocomplete?
Question 2: Is it locally presentable? If so, is there a reasonable "algebraic" presentation of the category?
Notes:
Tychonoff's theorem is what tells us that the category of compact Hausdorff spaces has products / that the category of commutative unital $C^\ast$-algebras has coproducts. The analog of Tychonoff's theorem for $\kappa$-Lindelöf spaces (and I believe equivalently for $\kappa$-Lindelöf Hausdorff spaces) if and only if $\kappa$ is a strongly compact cardinal. So the above questions trivially have negative answers unless $\kappa$ is strongly compact. Thus we should assume in the above that $\kappa$ is a strongly compact cardinal, and we are squarely in the realm of studying large cardinals.
Gelfand duality can be thought of as being analogous to the duality between affine schemes and commutative rings, or to Stone duality between profinite sets -- a "geometric" category is dual to a category which is "algebraic", and hence locally presentable.
