We give a definition for univalent morphisms in finitely complete that generalizes the aforementioned definitions and completely focuses on the aspects, characterizing it via representability of certain functors, which should remind the reader of concepts such as adjunctions or limits.
We then prove that in a locally cartesian closed (that is not necessarily presentable) univalence of a morphism is equivalent to the completeness of a certain segal object we construct out of the morphism, characterizing univalence via internal which had been considered in a strict setting by stenzel.
We present a new, category theoretic point of view on finite ramsey theory.
Our aims are as follows: (1) to define the category theoretic notions needed for the development of finite ramsey theory, (2) to state, in terms of these category theoretic notions, the general fundamental fundamental ramsey results (of which various concreteramsey results are special cases), and (3) to give self-contained proofs within the category theoretic framework of these general results.
We present a formalization of category theory for the future version of the agda standard library, in particular so that the library integrates well with the standard library of agda, as well as supporting as many of the modes of agdaas possible.
The formalization revealed a number of potential design choices, and we present, motivate and explain the ones we picked.
Category theory is a powerful tool for modeling phenomena and communicating results.
Monoids are framed in terms of agents acting on objects, sheaves are introduced with primary examples coming from geography, and colored operads are discussedin terms of their ability to model self-similarity.