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I’m wondering now how consistent is the naming in this neck of the words, especially transporter category and Frobenius category. The latter does not mean Frobenius category. We’re coming close here to the intricate world of fusion systems.
I’ve left a reference where I took the terminology from.
Have moved discussion of Properties out of the Definition-section.
Have replaced the tom Dieck-reference item with a hyperlinked version:
(which means I had to remove the experimental bibtex functionality, for the time being)
and I changed the wording around that reference:
The term “EI-category” is not actually in that book, is it? What is there, on p. 73, is the explicit observation that “endomorphisms are automorphisms” on the orbit category and its relatives.
So I have made the wording refer to that verbiage on that page now, instead. But let me know if I am missing something.
Does anyone have any nicer names for this concept? I find “EI” to be a confusing term, because “I” could stand for either “isomorphism” or “identity” (the latter being the defining property of one-way categories). An actual adjective would be more aesthetic as well.
It’s a pretty standard term, unfortunately. (Perhaps you meant to refer to gaunt categories for the other notion; they don’t have to be direct.)
Perhaps you meant to refer to gaunt categories for the other notion; they don’t have to be direct.
Oh, the nLab entry for one-way categories is a little misleading at the moment, because they’re defined on the page for direct categories; however, one-way categories do not have to be direct. Perhaps it would be better to give them their own page.
Yes, I realized actually gaunt is somewhat different, that means every isomorphism is an identity, but you mean every endomorphism is an identity.
I don’t think I’ve ever heard anyone talk about one-way categories that aren’t direct, but I guess the definition allows that. The terminology seems somewhat strange in that case, though; what’s “one-way” about a category with this sort that isn’t direct? In particular, it doesn’t have to be skeletal, does it?
The terminology seems somewhat strange in that case, though; what’s “one-way” about a category with this sort that isn’t direct? In particular, it doesn’t have to be skeletal, does it?
I agree: “one-way” doesn’t seem like an accurate description unless you also impose that the category is skeletal (in which case you have an acyclic category).
Add a characterisation of EI-categories as those categories satisfying a “Schröder–Bernstein theorem”. (I’m sure the observation is not new, but I don’t know a reference for it.)
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