Abstract

When formalizing mathematics in constructive type theories, or more practically in proof assistants such as Coq or Agda, one is often using setoids. In this note we consider two categories of setoids with equality on objects and show, within intensional Martin-Löf type theory, that they are isomorphic. Both categories are constructed from a fixed proof-irrelevant family F of setoids. The objects of the categories form the index setoid I of the family, whereas the definition of arrows differs. The first category has for arrows triples →F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\rightarrow \,F)$$\end{document} where f is an extensional function. Two such arrows are identified if appropriate composition with transportation maps makes them equal. In the second category the arrows are triples 2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2)$$\end{document} where R is a total functional relation between the subobjects F,F↪Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F, F \hookrightarrow \Sigma $$\end{document} of the setoid sum of the family. This category is simpler to use as the transportation maps disappear. Moreover we also show that the full image of a category along an E-functor into an E-category is a category.