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Acyclic types and epimorphisms in homotopy type theory

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Tom day Jong!

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It is well known that the epimorphisms in the category of sets are exactly the surjections. But what are the epimorphisms of types? Thinking of types as spaces, and following literature in algebraic topology, we characterise epimorphisms of types in homotopy type theory (HoTT) as so-called acyclic maps. An acyclic map is a map whose fibres are acyclic types, and a type is acyclic if its suspension is contractible. We also consider a weaker notion called k-acyclicity (for k ≥ -2) and show for example that the classifying type of a group G is 2-acyclic if and only if the group G is perfect, i.e. its abelianisation is trivial. I will not assume any familiarity with HoTT and pause to introduce the relevant notions.

This is joint work with Ulrik Buchholtz and Egbert Rijke.

This talk is part of the Theoretical computer science seminar series.

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