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Results tagged with homotopy-theory
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Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
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Categories enriched in $(m,n)\text{-Cat}$ with Crans–Gray tensor product
(All higher categories will be strict unless otherwise noted):
It is a somewhat folklorish fact that if you equip the category of $(m,n)$ categories with the cartesian monoidal structure, then a categ …
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Is the adjunction between spaces and chain complexes monadic?
Consider the adjunction of $\infty$-categories $\mathbb{Z}[-]: \mathcal{Spaces} \rightleftarrows \text{Ch}_{\ge 0}(\mathbb{Z}): |-|$ where the left adjoint takes a space to its singular chain complex …
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Simplicial objects in quasicategory which come from homotopy coherent nerve
Let $\mathcal{C}$ be a simplicially enriched category whose Hom-objects are all Kan complexes. Denote by $N\mathcal{C}$ the homotopy-coherent nerve of $\mathcal{C}$, which is a quasicategory. Suppose …
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Operads and Inverses
One slightly displeasing thing about the theory of operads is that they are unable to encode the structure of inverses; e.g. the recognition principle for $n$-fold loop spaces says "there is an operad …
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