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I am looking for a descriptive adjective to describe the following special property that some objects in some categories enjoy: their endomorphism monoids are groups. Of course, one way this could happen is if the ambient category is a groupoid, but in the cases I am interested in, the objects in question do receive and emit noninvertible morphisms; only their endomorphisms are invertible. For example, in the category of commutative rings, polynomial rings do not enjoy this property, but (if I'm not mistaken) fields of finite transcendence degree do.

Any suggestions on a term to use?

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    $\begingroup$ Since in the abelian context the role of such is played by simple objects - all of their nonzero endomorphisms are automorphisms - maybe call these simple too? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 4, 2022 at 19:28
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    $\begingroup$ Categories in which all objects have that property are sometimes called EI-categories. They are, in a way, the marriage of groupoids with posets. $\endgroup$
    – Tilman
    Commented Jan 4, 2022 at 19:38
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    $\begingroup$ Fields of finite transcendence degree enjoy what? the endomorphism of $\mathbf{Q}(t)$ mapping $P(t)$ to $P(t^2)$ is not invertible. $\endgroup$
    – YCor
    Commented Jan 4, 2022 at 19:55
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    $\begingroup$ @LSpice: rigid means (to me) that it has no automorphisms (probably no endomorphisms, but in the example I have in mind, all morphisms are mono anyway). How about corporeal, as a nod to the example of fields. I'm assuming this word hasn't been used before in category theory.) Another idea: incompressible. $\endgroup$
    – Paul Taylor
    Commented Jan 4, 2022 at 21:10
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    $\begingroup$ As pointed out by Tilman, it is pretty common to use EI-category to mean every Endomorphism in a category is Invertible. I am not sure if category theorists use this but people doing representations and cohomology of categories do and it is fairly widely used now. So I think EI-object would be reasonable. The EI-objects give the largest full subcategory which is an EI-category and the Endomorphism is Invertible acronym still makes sense. $\endgroup$
    – Benjamin Steinberg
    Commented Jan 5, 2022 at 15:14

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