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Categories can be used to model databases, and have been used for data integration. See also this paper. There's a pretty gentle introduction in chapter 3 of Seven Sketches in Compositionality.
Coalgebras are a generalization of automata, and have provided insight into notions of bisimulation. I saw a talk about this, but also found this paper
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Not what you are hoping for, but:
An early version of the OCaml language was based on the idea of a categorical abstract machine "represented by Cartesian closed category".
Haskell's monad classes "are based on the monad construct in category theory".
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A silly example: Checking if a graph has more than m nodes. :-)
Some more interesting examples:
Checking if a graph is connected.
Checking if a graph has an Eulerian cycle. A graph is Eulerian iff all degrees are even and the graph is connected: https://en.wikipedia.org/wiki/Eulerian_path
Checking if a graph has a perfect matching: https://en.wikipedia.org/...
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Specifically related to security, we can cite the problem when system administrators assess the impact of an access control policy on his system's security. In the paper "Protection in operating systems" Harrison, Ruzzo, and Ullman [1] presented a formal model of "access control". A state in this system is denoted by a set of objects, ...
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