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Say in a category, for any two objects $A,B$, we have an isomorphism $\iota_{AB}:A\to B$ with the property that $\iota_{BC}\circ\iota_{AB}=\iota_{AC}$ and $\iota_{AA}=\mathit{id}$.

Essentially, such a (family of) isomorphisms canonically identifies all objects in the category.

Is there a name for this property?

Examples:

  • In the category of singleton sets, there is a unique isomorphism between any two objects, it has this property.
  • In the category of one-dimensional vector spaces with designated bases, there is a unique isomorphism that maps a basis state to a basis state. It has this property. (This is the situation I am faced with and I want to use a standard term for this notion in my paper, if at all possible.)
  • In the category of totally ordered sets of a fixed finite cardinality $n$, the unique order preserving bijection between objects has this property.
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  • $\begingroup$ Doesn’t every category in which all objects are isomorphic have this property? Fix an object $X$, and for each object $A$ pick an isomorphism $\varphi_A: X\to A$. Then let $\iota_{AB}=\varphi_B\varphi_A^{-1}$. $\endgroup$
    – Jeremy Rickard
    Commented Sep 27, 2023 at 12:49
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    $\begingroup$ Ah, but reading the question more carefully, I think you’re asking for a name for the collection of isomorphisms, rather than a name for a category which has such a collection? $\endgroup$
    – Jeremy Rickard
    Commented Sep 27, 2023 at 12:54
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    $\begingroup$ Upon choosing an object, this feels like it's equivalent to the data of a fixed adjoint equivalence to the trivial category. $\endgroup$
    – Sean Sanford
    Commented Sep 27, 2023 at 13:25
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    $\begingroup$ I have never heard of a name for such a thing, but the first name that occurs to me is "wide clique". $\endgroup$
    – Mike Shulman
    Commented Sep 27, 2023 at 16:43
  • $\begingroup$ @JeremyRickard Yes, I am trying to name the property. Basically, I would write a sentence such as "we show that $\iota_{AB}$ is XXX, that is satisfies ...". If there exists no term, that's also ok, then I'll make up something, but I wanted to use an existing term if there is one. $\endgroup$
    – Dominique Unruh
    Commented Sep 28, 2023 at 13:10

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