Eilenberg-MacLane Spaces in Homotopy Type Theory
CSL-LICS ‘14 66 (2014) 1-9
on Eilenberg-MacLane spaces in homotopy type theory.
Abstract. Homotopy type theory is an extension of Martin-Löf 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 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.
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