On branched covers in topos theory.
We show that for an arbitrary Set-endofunctor T the generalized membership function given by a sub-cartesian transformation μ from T to the filter functor 𝔽 can be alternatively defined by the collection of subcoalgebras of constant T-coalgebras. Sub-natural transformations ε between any two functors S and T are shown to be sub-cartesian if and only if they respect μ. The class of T-coalgebras whose structure map factors through ε is shown to be a covariety if ε is a natural and sub-cartesian mono-transformation....
In this paper, we use a characterization of -modules such that to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting to be the local cohomology functor of with respect to the maximal ideal where is the Krull dimension of .