Towards a Topological Model of Homotopy Type Theory
The model of homotopy type theory in simplicial sets  has proven to be a grounding and motivating influence in the development of homotopy type theory. The classical theory of topological spaces has also proven to be motivational to the subject. Though the Quillen equivalence between simplicial sets and topological spaces provides, in some weak sense, a model in topological spaces, we explore the extent to which the category of topological spaces may be a more direct and strict model of homotopy type theory. We define a notion of model of homotopy type theory, and show that the category of topological spaces fully embeds into such a model.
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