A metaphor for categories and mappings
This section in particular:
For a really simple, real world example, consider a lawnmower. When out of gas/petrol, it needs to be refueled. The category of empty lawnmowers has an arrow/mapping/morphism to the category of full lawnmowers. Here the focus is more on the _process_ of mapping one category into another, or mapping one collection of things into another collection.
It's been said that category theory places mappings/arrows on a fully-equal footing with objects/points/things.
Now back to the lawnmower example. A child walks out to watch his father mowing the lawn. The lawmower runs out of gas. The child says "Daddy, I think the lawnmower needs a drink."
What the child is doing is using his own set of mappings, from thirsty to not thirsty, to metaphorize the process of fueling the lawnmower. There's a mapping between these mappings, a functor. (I happen to think this categorical way of thinking is immensely important for how we understand the world, how we might program artificial intelligences, and so on. Even more than "design patterns," I think we and other creatures chunk the world into pieces we can understand by finding the mappings and functors and so on which allow us to grok the world. We are, I think, category theory engines.)
Also this in the comments
A more useful way to think about category theory compared to set theory is to view it as a kind of inversion of reasoning. In set theory everything is built up, and we understand objects by seeing what they are made of -- picking apart their internals. We relate two objects by comparing what they are made of. Category theory turns that on its head and takes relationships between objects as primitive. From CT's point of view objects are opaque; we understand an object not by peeking inside at it's internal structure, as we would if we were using set theoretic thinking, but rather by examining how the object relates to other objects.
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