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Fundamental groupoids of k-graphs
k-graphs are higher-rank analogues of directed graphs which were first
developed to provide combinatorial models for operator algebras of
Cuntz-Krieger type. Here we develop a theory of the fundamental groupoid of a
k-graph, and relate it to the fundamental groupoid of an associated graph
called the 1-skeleton. We also explore the failure, in general, of k-graphs to
faithfully embed into their fundamental groupoids.Comment: 12 page
Modalities in homotopy type theory
Univalent homotopy type theory (HoTT) may be seen as a language for the
category of -groupoids. It is being developed as a new foundation for
mathematics and as an internal language for (elementary) higher toposes. We
develop the theory of factorization systems, reflective subuniverses, and
modalities in homotopy type theory, including their construction using a
"localization" higher inductive type. This produces in particular the
(-connected, -truncated) factorization system as well as internal
presentations of subtoposes, through lex modalities. We also develop the
semantics of these constructions
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