Timeline for Is there a name for objects all of whose endomorphisms are automorphisms?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jan 5, 2022 at 16:10 | comment | added | Paul Taylor | If "Ei" is the German word, then every endomorphism is indeed either an isomorphism or a zero! | |
| Jan 5, 2022 at 15:23 | comment | added | Theo Johnson-Freyd | @PaulTaylor I completely agree. In categories of spaces, this condition typically only holds for points. So I was tempted by something like "pointal"... | |
| Jan 5, 2022 at 15:14 | comment | added | Benjamin Steinberg | 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. | |
| Jan 5, 2022 at 14:14 | comment | added | Paul Taylor | You need some more concrete examples to suggest a suitable name. The one I'm thinking of is sets ($\in$-structures) considered as extensional well founded $\mathcal P$-coalgebras, for which see section 7 of (paultaylor.eu/ordinals/welfcr.pdf) | |
| Jan 5, 2022 at 13:53 | comment | added | Ville Salo | Often "coalescence" in topological dynamical systems is discussed in the context of minimal systems, where trivially every endomorphism is epic. | |
| Jan 5, 2022 at 13:50 | comment | added | Ville Salo | Not sure this is helpful, but you can split this into "every endomorphism is epic" and "every epic endo is auto", and the latter is/could be called "Hopfian". | |
| Jan 5, 2022 at 13:48 | comment | added | Theo Johnson-Freyd | @PaulTaylor I would interpret "incompressible" to mean that any morphism emitted by the object is a monomorphism. Or maybe (more strongly) every morphism emitted is an injection, which makes sense if my category is concrete (= consists of sets+structure). | |
| Jan 5, 2022 at 13:38 | comment | added | Theo Johnson-Freyd | @YCor Oops, wasn't thinking straight. | |
| Jan 5, 2022 at 13:36 | history | edited | Theo Johnson-Freyd | CC BY-SA 4.0 |
Oops
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| Jan 4, 2022 at 21:10 | comment | added | Paul Taylor | @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. | |
| Jan 4, 2022 at 19:55 | comment | added | YCor | Fields of finite transcendence degree enjoy what? the endomorphism of $\mathbf{Q}(t)$ mapping $P(t)$ to $P(t^2)$ is not invertible. | |
| Jan 4, 2022 at 19:38 | comment | added | Tilman | Categories in which all objects have that property are sometimes called EI-categories. They are, in a way, the marriage of groupoids with posets. | |
| Jan 4, 2022 at 19:28 | comment | added | მამუკა ჯიბლაძე | 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? | |
| Jan 4, 2022 at 19:23 | comment | added | Robin Tucker-Drob | This is sometimes called "coalescence" in dynamical systems. The term is used in the measured context here: arxiv.org/abs/1808.00341 and in the topological context here: arxiv.org/abs/1103.2647 | |
| Jan 4, 2022 at 19:00 | comment | added | LSpice | This makes me think of 'rigid', but I guess that, as usually understood, that is the related (?) but distinct property that the only automorphism is the identity. | |
| Jan 4, 2022 at 18:55 | history | asked | Theo Johnson-Freyd | CC BY-SA 4.0 |