Basic Concepts on Higher Algebra
Algebraic Pattern Algebraic pattern is a blueprint for a notion of functors on a fixed category satisfying a Segal condition, suitable for formalizing homotopy-coherent algebra in the Cartesian setting. Informally, algebraic pattern generalizes the active and inert morphisms in operads and chooses certain objects to control the Segal condition. Definition 1. An algebraic pattern is a category $\mathcal{O}$ equipped with: A collection of objects called elementary objects. A factorization system $(\mathcal{O}^{\text{inv}}, \mathcal{O}^{\text{act}})$ where every morphism factors uniquely (up to equivalence) as an inert morphism followed by an active morphism. We let $\mathcal{O}^{\mathrm{el}}$ denote the full subcategory of $\mathcal{O}$ spanned by the elementary objects and the inert morphisms between them. For any object $X \in \mathcal{O}$, we also write ...