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 homotopytypetheory.org
Localization as an Inductive Definition | Homotopy Type Theory
| | www.jeremykun.com
Functoriality | Math Programming
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| | Last time we worked through some basic examples of universal properties, specifically singling out quotients, products, and coproducts. There are many many more universal properties that we will mention as we encounter them, but there is one crucial topic in category theory that we have only hinted at: functoriality. As we've repeatedly stressed, the meat of category theory is in the morphisms. One natural question one might ask is, what notion of morphism is there between categories themselves?
| | math.andrej.com
Mathematics and Computation | The elements of an inductive type
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| | www.twanvl.nl
Type theory with indexed equality - the theory
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| | xorshammer.com
A Logical Interpretation of Some Bits of Topology - XOR's Hammer
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| Edit: These ideas are also discussed here and here (thanks to Qiaochu Yuan: I found out about those links by him linking back to this post). Although topology is usually motivated as a study of spatial structures, you can interpret topological spaces as being a particular type of logic, and give a purely logical, non-spatial...