My blog post for the Applied Category Theory Adjoint School (jww Rubén Maldonado), is now up on the n-Category Café: https://golem.ph.utexas.edu/category/2024/08/prismatic_category_theory.html#more
In this blog post, we explore properties of Grothendieck fibrations, with an eye toward getting a better handle on higher-categorical notions such as double fibrations. For this purpose, we develop a graphical framework for displayed categories and fibrations, and subsequently show how this framework can be interpreted in the internal language of simplicial sets, which paves the way for using simplicial type theory as a tool in the study of double categories, double fibrations, etc!
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