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。组合模型一直是特别重要的，现在已经渗透到了这个主题中。更广泛地说，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产生--更高代数K-变种理论的提出是一个早期的、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，上同调下降在20世纪80年代末单纯前垂直波的局部同伦理论中被形式化。局部同伦理论是理据同伦理论的基础，并为堆栈理论提供了概念语境。在定义椭圆上同调理论和拓扑模形式时，形式群律的堆栈上出现了大量的谱。这些想法中的许多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.