we start this article by rebuilding higher operads of weak higher transformations, and correct those in cite{cambat}. as in cite{cambat} we propose an operadic approach for weak higher $n$-transformations, for each $ninmathbb{n}$, where such weak higher $n$-transformations are seen as algebras for specific contractible higher operads. the last chapter of this article asserts that, up to precise...

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental ngroupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applicatio...

We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricateg...

We introduce flagged (∞, n)-categories and prove that they are equivalent to Segal sheaves on Joyal’s category Θn. As such, flagged (∞, n)-categories provide a model-independent formulation of Segal sheaves. This result generalizes the statement that n-groupoid objects in spaces are effective, as we explain and contextualize. Along the way, we establish a useful expression for the univalent-com...