Applied homotopy theory

同伦理论的应用形成了这门学科最引人注目的方面。组合模型一直是特别重要的，现在渗透的主题。更一般地说，单纯结构存在于整个代数几何中，同伦类型以许多方式从几何对象中产生-高等代数簇的K-理论的表述是一个早期的，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，这在1980年代后期的单纯预层的局部同伦理论中得到了形式化。局部同伦理论是动机同伦理论的基础，并为堆栈理论提供了概念背景。在定义椭圆上同调理论和拓扑模形式时，出现了形式群律栈上的谱的预层。许多这些想法在数学物理和计算机科学中有应用。

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.