Filter categories preserving desirable signal characteristics

This paper considers the problem of characterizing nonlinear digital filters via their qualitative input-output behavior. For example, it is known that (non-causal) symmetric median filters preserve arbitrary monotone sequences. Behavioral characterizations ask converse questions like: what can be said about the class of all filters F that preserve monotonicity? The primary point of this paper is to illustrate that category theory provides a natural framework for examining such questions.