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A surprising story of how a computer was taught to prove some theorems in finitely complete categories

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Notes for the talk given at the 2021 Congress of South African Mathematical Society on 29 November.  1. Finitely complete categories A finitely complete category is a category that has finite products and equalizers (and hence, all finite limits). Not every category is finitely complete, but most categories of mathematical structures are.  There is a representation theorem for finitely complete categories (Yoneda embedding), which allows to present any category as a full subcategory of a (larger) category of presheaves of sets, which is closed under all limits that exist in the category. This means that a lot of times, proving a theorem in a finitely complete category involving finite limits, reduces to proving the same theorem in the category of sets.  For instance, the fact that the product of objects is commutative, up to a canonical isomorphism, can be deduced from the fact that the same is true for the cartesian product of sets. Or, the fact that the composite of two monomorphisms