Internal categories in crossed semimodules and Schreier internal categories

Internal Algebra Classifiers as Codescent Objects of Crossed Internal Categories

Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects ari...

Internal Categories and Quantum Groups

Enriched Categories , Internal Categories and Change of Base Dominic Verity

Weak complicial sets and internal quasi-categories

It is well known that we may represent (strict) ω-categories as simplicial sets, via Street’s ω-categorical nerve construction [2]. What may be less well known, is that we may extend Street’s nerve functor to one which has been shown to be fully-faithful (Verity [3]). This is achieved by augmenting each simplicial set in the codomain of this functor with a specified subset of thin simplices and...

Internal Languages for Autonomous and *-Autonomous Categories

We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and-autonomous categories, in the same sense that the simply-typed-calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgements are presented in the style of Gentzen's Sequent Calculus. A notable feature is the systematic tr...