Homological and homotopical algebra are two powerful branches of mathematics with deep connections and applications across various fields, including algebraic topology, algebraic geometry, and mathematical physics. While they originate from distinct mathematical backgrounds, they share a common goal of studying algebraic structures through the lens of algebraic topology and category theory. In this article, we explore the fundamentals of homological and homotopical algebra, highlighting their similarities, differences, and the bridges that connect them.
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