Eilenberg-MacLane Spaces in Homotopy Type Theory

ABSTRACT. Homotopy type theory is an extension of Martin-L\"of type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs of homotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this paper, we give a computer-checked construction of \emph{Eilenberg-MacLane spaces}. For an abelian group $G$, an Eilenberg-MacLane space $K(G,n)$ is a space (type) whose $n^{th}$ homotopy group is G, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in algebraic topology; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of cohomology with coefficients in $G$. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.