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Let $\mathcal{K}$ be a $2$-category. Is there a special name of those objects $B \in \mathcal{K}$ which have the property that the category $\mathrm{Hom}_{\mathcal{K}}(B,C)$ is essentially discrete for all $C \in \mathcal{K}$? This means that for every two morphisms $f,g : B \to C$ any $2$-morphism $f \to g$ is an isomorphism, and it is unique if it exists.

Also, is there some intuition what this property actually means, intuitively?

Assuming that $\mathcal{K}$ has more properties (such as the existence of certain limits), can we simplify the property perhaps to some "internal" information of $B$?

Here is the example which motivates my question: Consider the $2$-category $\mathrm{Cat}_{c\otimes/R}$ of cocomplete $R$-linear tensor categories and let $A$ be a commutative $R$-algebra. Then $\mathrm{Mod}(A) \in \mathrm{Cat}_{c\otimes/R}$ has this property, since $\mathrm{Hom}_{c\otimes/R}(\mathrm{Mod}(A),\mathcal{C}) \simeq \mathrm{Hom}_R(A,\mathrm{End}(1_\mathcal{C}))$ and the latter is a set. I have recently proven that $\mathrm{Qcoh}(X)$ has this property as well, where $X$ is any quasi-compact quasi-separated $R$-scheme. When $X$ is an algebraic stack, usually $\mathrm{Qcoh}(X)$ doesn't have this property.

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I would call those objects "codiscrete" or "co-0-truncated", since "discrete" and "0-truncated" are used for the dual property, e.g. here and here. It's equivalent to saying that $B$ is equivalent to the copower $B \odot (\cdot \rightrightarrows \cdot)$.

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  • $\begingroup$ Thanks, Mike! In fact I looked for "discrete object" at the nlab, but only saw the "wrong" page: ncatlab.org/nlab/show/discrete+object $\endgroup$ – Martin Brandenburg Jan 10 '20 at 19:09
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    $\begingroup$ @MartinBrandenburg Note that that page links to the correct page in the second paragrah of the "Idea" section. $\endgroup$ – Mike Shulman Jan 11 '20 at 20:26

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