IdentityFunctor 2007-01-24 23:28:08 2007-10-24 00:34:11 Definition functor \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} \usepackage{psfrag} % define commands here Let $\mathcal{C}$ be a category. The \emph{identity functor} of $\mathcal{C}$ is the unique functor, written $I_{\mathcal{C}}$, such that for every object $A$ and every morphism $\alpha$ in $\mathcal{C}$, we have $$I_{\mathcal{C}}(A)=A\quad\mbox{ and }\quad I_{\mathcal{C}}(\alpha)=\alpha.$$ To verify that $I_{\mathcal{C}}$ is indeed a functor, we note that $I_{\mathcal{C}}(1_A)=1_A=1_{I_{\mathcal{C}}(A)}$, where $1_A$ is the identity morphism of $A$, and $I_{\mathcal{C}}(\alpha\circ\beta)=\alpha\circ \beta=I_{\mathcal{C}}(\alpha)\circ I_{\mathcal{C}}(\beta)$. For any functor $F:\mathcal{C}\to \mathcal{D}$, we have $F\circ I_{\mathcal{C}}= I_{\mathcal{D}}\circ F=F$. Since every category gives rise to its unique identity functor, we can think of \emph{the identity functor} $I$ as a (covariant) functor on \textbf{Cat}, the category of (small) categories. It is given by taking any category $\mathcal{C}$ to itself and any functor $F:\mathcal{C}\to \mathcal{D}$ to itself.