Category Theory For Beginners: Everyday Language
In this video I discuss how to represent everyday language using category theory. The idea behind this approach (ontology logs) is simple, but extremely powerful. The basic idea is to have objects representing indefinite noun phrases and arrows representing functional relationships. This approach allows us to encode ideas from everyday language (English language in this case) into concrete mathematical statements that can be represented in category theory. For example, we describe how the notion `a mother is older than her child' can be encoded in category theory. I also describe how limits like products, equalizers, and pullbacks (and also colimits like coproducts) occur in simple cases involving everyday language. I also show how ontology logs can be used to represent computations.
In the last scene I use recursion to represent a function h which takes an input (n,x) and outputs (0,f^n (x)) where f^n (x) is the element of S obtained by taking x and applying f to it n times. This can be seen as follows. We call the bottom left inclusion function in0. We call the right inclusion function in1. We call the top inclusion function in2. When n is greater than 0 we can think of (n,x) as belonging to the top left type. Since (h after in2)=k, we have
(h after in2)((n,x))= h((n,x))= (k((n,x)) = h((n-1,f(x))
in this case. Also, since (h after in0)=in1 we have
(h after in0 ((n,x))=h((0,x))=in1((0,x))=(0,x). So if n is greater than zero than h((n,x))= h((n-1,f(x)). Also h((0,x))=(0,x). So when we start with a positive n, we have that h keeps applying f to the right entry, while reducing the left entry by one. This continues until the left entry is zero, when it outputs the result.
For more on ontology logs (ologs) you can follow the links below to Spivak and Kent's paper on ontology logs, or Ryan Wisnesky's dissertation on Functional Query Languages with Categorical Types.
https://arxiv.org/pdf/1102.1889.pdf
https://dash.harvard.edu/bitstream/ha...
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