Category is a most abstract mathematical notion, which, however, found its way to computer science and physics, for example. Here is an attempt to present the category theory for a broader audience.
Abstract:
There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be applied throughout the sciences as a framework for modeling phenomena and communicating results. In order to target the scientific audience, this book is example-based rather than proof-based. For example, monoids are framed in terms of agents acting on objects, sheaves are introduced with primary examples coming from geography, and colored operads are discussed in terms of their ability to model self-similarity.
Mathematical Biology 
This seems really interesting, I will look through it. My supervisor always reasons with category theory, so I’ve picked up scraps of it and how it applies to computer science (and a little bit to physics). It is definitely a great way to talk more clearly, and to understand what problems you are addressing. It would be nice to see it used in modeling heavy fields. Have you seen much use of category theory in biology?
No, I did not see any speculations why categories might be of some use in mathematical models in biology. However, I definitely do expect to see some quite soon, just because of the fact that if something can be done it will be done 🙂