Eilenberg-MacLane Spaces in Homotopy Type Theory Daniel R. Licata ∗ Eric Finster Wesleyan University Inria Paris Rocquencourt
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[email protected] Abstract using the homotopy structure of types, and type theory is used to Homotopy type theory is an extension of Martin-Löf type theory investigate them—type theory is used as a logic of homotopy the- with principles inspired by category theory and homotopy theory. ory. In homotopy theory, one studies topological spaces by way of With these extensions, type theory can be used to construct proofs their points, paths (between points), homotopies (paths or continu- of homotopy-theoretic theorems, in a way that is very amenable ous deformations between paths), homotopies between homotopies to computer-checked proofs in proof assistants such as Coq and (paths between paths between paths), and so on. In type theory, a Agda. In this paper, we give a computer-checked construction of space corresponds to a type A. Points of a space correspond to el- Eilenberg-MacLane spaces. For an abelian group G, an Eilenberg- ements a;b : A. Paths in a space are modeled by elements of the identity type (propositional equality), which we notate p : a = b. MacLane space K(G;n) is a space (type) whose nth homotopy A Homotopies between paths p and q correspond to elements of the group is G, and whose homotopy groups are trivial otherwise. iterated identity type p = q. The rules for the identity type al- These spaces are a basic tool in algebraic topology; for example, a=Ab low one to define the operations on paths that are considered in they can be used to build spaces with specified homotopy groups, homotopy theory.