We develop foundations for the category theory of $\infty$-categories parametrized by a base $\infty$-category. Our main contribution is indexed homotopy limits and colimits, which specializes to $G$-colimits $G$ finite group when chosen be orbit $G$. apply this show that $G$-$\infty$-category $G$-spaces freely generated under contractible $G$-space, thereby affirming conjecture Mike Hill.

“Mathematical chemistry” as an academic field is said to have been proposed by Weyl in the 20th century a way of thinking that abstracts and expresses “variables”, “symbols” “functions” [...]

following the categorical approach to universal algebra through algebraic theories, proposed by f.~w.~lawvere in his phd thesis, this paper aims at introducing a similar setting for general topology. the cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. we introduce t...

This article presents a development of Category Theory in Isabelle. A Category is defined using records and locales in Isabelle/HOL. Functors and Natural Transformations are also defined. The main result that has been formalized is that the Yoneda functor is a full and faithful embedding. We also formalize the completeness of many sorted monadic equational logic. Extensive use is made of the HO...

Galois theory translates questions about fields into questions about groups. The fundamental theorem of Galois theory states that there is a bijection between the intermediate fields of a field extension and the subgroups of the corresponding Galois group. After a basic introduction to category and Galois theory, this project recasts the fundamental theorem of Galois theory using categorical la...

Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number of such “set-theore...