How do I contain thee?

Invariants are an important tool for the verification of complex systems. One very simple class of invariants for compositional reasoning is given by linear constraints on the numbers of subword occurrences. This class of properties exhibits some quite interesting connections to mathematical fields such as topology. I have recently been interested in generalizing these concepts to more general structures than words, and investigate properties given in terms of subobject counters in suitably nice categories. In this talk, I will give an overview of this work in progress.