## 83.1 Introduction

This chapter develops some theory concerning simplicial topological spaces, simplicial ringed spaces, simplicial schemes, and simplicial algebraic spaces. The theory of simplicial spaces sometimes allows one to prove local to global principles which appear difficult to prove in other ways. Some example applications can be found in the papers [faltings_finiteness], [Kiehl], and [HodgeIII].

We assume throughout that the reader is familiar with the basic concepts and results of the chapter Simplicial Methods, see Simplicial, Section 14.1. In particular, we continue to write $X$ and not $X_\bullet $ for a simplicial object.

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