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The ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions. However, continuous distributions have to be discretized because the corresponding measures cannot be defined on all subsets of their carriers. This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory. We study the initial $sigma$-frame and the corresponding induced topology on arbitrary sets. Based on these intrinsic topologies we define valuations and lower integrals on sets, and pro
This is an introduction to Homotopy Type Theory and Univalent Foundations for philosophers, written as a chapter for the book Categories for the Working Philosopher (ed. Elaine Landry)
We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order
We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational sta
We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program lo