I am a computer scientist with a CS degree from the 80's. I am learning category theory by myself. I am looking for advice about learning category theory. E.g. it is quite helpful in learning category theory to have examples from a familiar topic like logic or programming languages.
What I know
I have read the answers to this question and the paper Physics, Topology, Logic and Computation: A Rosetta Stone which builds up concepts with parallels between category theory, logic, and theory of computation.
I know about the following categories:
• monoidal categories
• braided monoidal categories
• closed monoidal categories
• symmetric monoidal categories
• closed braided monoidal categories
• compact monoidal categories
• cartesian categories
• closed symmetric monoidal categories
• compact braided monoidal categories
• cartesian closed categories
• compact symmetric monoidal categories
Are there any other major categories that a computer scientist should know about?
Were there any that have fallen out of favor? If yes, why? Have they been replaced with other categories?
When thinking about categories should I focus on the arrows first before the objects?