Simplicial
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1 Introduction
Simplicial things come up in mathematics a lot. There are at least three related places of origin. The first is homotopy theory. Simplicial sets give a way of modeling the homotopy theory of topological spaces, and are often a convenient model for higher categories. The second is in homological algebra. There is the Dold Kan correspondence which identifies homotopy theories of chain complexes and simplicial objects, which suggests that simplicial objects can be useful for applying homotopical methods in nonabelian settings. The third place it arises is when studying monoids. Namely, the simplex category classifies monoids in a tensor category, meaning that tensor functors from \(\Delta \) coincide with monoid objects. This can be used in the construction of canonical resolutions from (co)monads, and in the formalism of monoidal \(\infty \)-categories.
The following are sources for this document:
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• Goerss, Jardine - Simplicial Homotopy Theory
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• Riehl - Categorical Homotopy Theory
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• Lurie - Higher Topos Theory
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• Lawson - Localization of Enriched Categories and Cubical Sets
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• Jardine - Categorical Homotopy Theory
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• Dwyer, Kan - Simplicial Localization of Categories