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Homediscrete category
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discrete category
A category $\mathcal{C}$ is said to be a discrete category if the only morphisms in $\mathcal{C}$ are the identity morphisms associated with each of the objects in $\mathcal{C}$.
For example, every set can be regarded as a discrete category. The objects are just the elements of the set. Furthermore, $\hom(a,a)$ is identified with $\{a\}$, and $\hom(a,b)=\varnothing$ if $a\neq b$.
Remarks.

A discrete category with one object is called a trivial category. For every category $\mathcal{C}$, there is only one functor from $\mathcal{C}$ to a trivial category. Hence, any trivial category is a terminal object in Cat, the category of small categories.

A discrete category with no objects is called the empty category. For every category $\mathcal{C}$, there is only one functor from the empty category to $\mathcal{C}$. This functor is called the empty functor, where both the object and morphism functions are the empty set $\varnothing$. Thus, the empty category is the initial object in Cat.

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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