Pure morphisms in pro-categories

AbstractPure epimorphisms in categories pro-C, which essentially are just inverse limits of split epimorphisms in C, were recently studied by J. Dydak and F.R.R. del Portal in connection with Borsuk’s problem of descending chains of retracts of ANRs. We prove that pure epimorphisms are regular epimorphisms whenever C has weak finite limits, or pullbacks, or copowers. This improves the results of the above paper, and the results of the present authors on pure subobjects in accessible categories. We also turn to pure monomorphisms in pro-C, essentially just inverse limits of split monomorphisms in C, and prove that they are regular monomorphisms whenever C has finite products or pushouts

Discrete equational theories

We introduce discrete equational theories where operations are induced by those having discrete arities. We characterize the corresponding monads as monads preserving surjections. Using it, we prove Birkhoff type theorems for categories of algebras of discrete theories. This extends known results from metric spaces to general symmetric monoidal closed categories.Comment: 13 page