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| A category $\mathcal{C}$ is said to be a \emph{discrete category} if the only morphisms in $\mathcal{C}$ are the identity morphisms associated with each of the objects in $\mathcal{C}$. |
A category $\mathcal{C}$ is said to be a \emph{discrete category} if the only morphisms in $\mathcal{C}$ are the identity morphisms associated with each of the objects in $\mathcal{C}$. |
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| A discrete category with one object is called a \emph{trivial category}. A discrete category with no objects is called the \emph{empty category}. |
Given any category $\mathcal{C}$, the smallest subcategory consisting of all objects in $\mathcal{C}$ is discrete. |
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| \textbf{Remark}. Given any category $\mathcal{C}$, the smallest subcategory consisting of all objects in $\mathcal{C}$ is discrete, which is also the largest discrete subcategory in $\mathcal{C}$ (largest in the sense that it contains all objects of $\mathcal{C}$). For every object $X\in \mathcal{C}$, we can associate the trivial category $\mathcal{C}_X$ consisting of one object, $X$, and one morphism $1_X$. |
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