Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Handbook of logic in computer science
A proof-irrelevant model of Martin-Löf's logical framework
Mathematical Structures in Computer Science
Modular correspondence between dependent type theories and categories including pretopoi and topoi
Mathematical Structures in Computer Science
Quotients over Minimal Type Theory
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
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Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with universes that are as small as possible (and thus, the type theory is as weak as possible). In particular, we will consider epimorphisms and disjoint sums. We show that, given the minimal type universe, all epimorphisms are surjections, and disjoint sums exist. Further, without universes, there are countermodels for these statements, and if we use the Logical Framework formulation of type theory, these statements are provably non-derivable.