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Co- And Contravariant Model Structure
In higher category theory in mathematics, co- and contravariant model structures are special model structures on slice categories of the category of simplicial sets. On them, postcomposition and pullbacks (due to its application in algebraic geometry also known as base change) induce adjoint functors, which with the model structures can even become Quillen adjunctions. Definition Let A be a simplicial set, then there is a slice category \mathbf/A. With the choice of a model structure on \mathbf, for example the Joyal or Kan–Quillen model structure, it induces a model structure on \mathbf/A. * ''Covariant cofibrations'' are monomorphisms. ''Covariant fibrant objects'' are the left fibrant objects over A. ''Covariant fibrations'' between two such left fibrant objects over A are exactly the left fibrations. * ''Contravariant cofibrations'' are monomorphisms. ''Contravariant fibrant objects'' are the right fibrant objects over A. ''Contravariant fibrations'' between two such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Category Theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic Invariant (mathematics), invariants of topological space, spaces, such as the Fundamental groupoid, fundamental . In higher category theory, the concept of higher categorical structures, such as (), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space. Strict higher categories An ordinary category (m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |