There are several notations and expressions commonly used in category theory to describe problems and their solutions. One common expression is to say that a category has a particular property, such as being complete, cocomplete, or Cartesian closed. Another common expression is to say that a category has limits or colimits, which are constructions that allow you to combine objects and morphisms in certain ways.

In terms of specific examples of how category theory has been applied in computer science and programming, there are several areas where it has been used:

Functional programming: Category theory has been used to study programming languages and their semantics, and to develop programming language features such as type systems and monads. Functional programming languages such as Haskell and Scala are based on concepts from category theory.

Database theory: Category theory has been used to define and reason about query languages and database schemas. One example of this is the Categorical Query Language (CQL), which is based on category theory and allows for expressive queries over databases.

Computer vision: Category theory has been used to develop algorithms for image processing and computer vision. One example of this is the development of a category-theoretic framework for image segmentation, which allows for a more principled and scalable approach to image segmentation.

A real-world success story for the application of principles of category theory is the development of the algebraic theory of databases. This theory is based on category theory and provides a foundation for reasoning about database schemas and queries. It has been used in the development of the CQL query language and has influenced the design of database systems such as MonetDB and PostgreSQL.
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