Applied homotopy theory

同伦理论的应用构成了这门学科最引人注目的方面。组合模型一直以来都是特别重要的，现在已经渗透到整个学科中。更一般地说，简单结构存在于整个代数几何中，同伦类型以多种方式从几何对象中产生——高代数k -变体理论的表述是一个早期的、具有挑衅性的例子。与k理论相关的计算困难导致了上同调下降的引入，它在20世纪80年代后期被形式化在简单预束的局部同伦理论中。局部同伦理论是动力同伦理论的基础，并为堆栈理论提供了一个概念背景。在定义椭圆上同调理论和拓扑模形式时，出现了形式群律叠加上谱的预捆。其中许多想法在数学物理和计算机科学中都有应用。

The applications of homotopy theory form the most compelling aspects of the subject. Combinatorial models have always been particularly important, and now pervade the subject. More generally, simplicial structures exist throughout algebraic geometry, and homotopy types arise from geometric objects in many ways - the formulation of the higher algebraic K-theory of varieties was an early, provocative example. Calculational difficulties associated with K-theory led to the introduction of cohomological descent, which was formalized in the local homotopy theory of simplicial presheaves in the late 1980s. Local homotopy theory is the basis for motivic homotopy theory, and gives a conceptual context for the theory of stacks. Presheaves of spectra on the stack of formal group laws have emerged in defining elliptic cohomology theories and topological modular forms. Many of these ideas have applications in mathematical physics and computer science.