# A Handbook of Model CategoriesCategories

A Handbook of Model Categories: Categories [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Handbook of Model CategoriesCategories

Part of the Algebra and Applications Book Series (volume 27)
16 pages

/lp/springer-journals/a-handbook-of-model-categories-categories-ckfj109Tmz
Publisher
Springer International Publishing
© Springer Nature Switzerland AG 2021
ISBN
978-3-030-75034-3
Pages
161 –177
DOI
10.1007/978-3-030-75035-0_9
Publisher site
See Chapter on Publisher Site

### Abstract

[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