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[In this chapter we will be equipping the category Cat\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Cat} $$\end{document}, of small categories and functors between them, with various model structures. This category admits all small limits and colimits, and moreover admits a Cartesian closed monoidal structure where the exponential objects are given by the functor categories. We will consider the restriction of these model structures to the full subcategories of posets and groupoids which we denote Grpd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Grpd} $$\end{document} and Pos\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Pos} $$\end{document}, respectively.]
Published: Oct 30, 2021
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