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Created left cancellative category. This is a useful technical term.
One natural (and non-posetal) example is the category of fields with ring homomorphisms as the morphisms—provided that the zero ring is removed from it.
Incidentally: is there a usual technical term for, when considering any category $\mathsf{C}$, the full subcategory obtained by removing all terminal objects of $\mathsf{C}$?
Who says the zero ring is a field?
All the sources I know of have $0 \neq 1$ as one of the field axioms.
Incidentally, the link for your first reference goes to a paper whose title is different from the one you name.
Thanks, viewed this way, there is indeed nothing to be removed. Yes, usually, the zero ring is not considered a field. Simplified the nlab entry accordingly.
Thanks also for mentioning that the title of the second reference had accidentally been used for the first reference, too. Corrected.
On removing terminal objects: the full subcategory of a category $\mathsf{C}$ obtained by removing a single terminal object $\bullet$ is isomorphic to the slice-category $\mathsf{C}/\bullet$ over $\bullet$.
In view of that, one can take the view that there is no need for a separate term.
However, to express removing all terminal objects in terms of the concept of slice-category only, one needs to repeatedly take slice-categories.
Moreover, a category can have a proper class of terminal objects.
What do you consider a usual reference for the operation of iteratively taking slice-categories, possibly for each object in a proper class of objects?
I’m a little confused by the claim
the full subcategory of a category $\mathsf{C}$ obtained by removing a single terminal object $\bullet$ is isomorphic to the slice-category $\mathsf{C}/\bullet$ over $\bullet$.
Isn’t $\mathsf{C}$ canonically isomorphic to $\mathsf{C}/\bullet$? (Put another way: what happens to the terminal objects $\bullet \to \bullet$ of the slice categories $\mathsf{C}/\bullet$ in this iterative process where you want to delete all terminal objects?)
@jesse correct. Also: any slice category has a canonical representative of terminal object, namely the identity arrow. Removing a single object from a category is an act of violence, though it can be done. Unless you also remove all the objects isomorphic to it, you still end up with something equivalent to the starting category.
I added poset and preorder to the list of tautological examples, and removed the one about subobjects of a given object being a non-tautological example, since that is a preorder. I added another non-tautological example: the category of nontrivial vector spaces equipped with nondegenerate inner products.
[never mind]
@jesse @DavidRoberts Thanks for the comments. Yes, (thinking about) removing (or adding) terminal objects violates the principle of equivalence, so should perhaps be avoided unless there is a necessity to do so.
@ Todd_Trimble: Thanks for the edits, they appear to improve the page.
Is the category of projective spaces an example?
Also, the category of probability spaces and measure preserving maps is right cancellative. (Or at least it morally should be right cancellative, because the image of each map should be measure $1$. To make it actually right cancellative you have to quotient together two maps if they differ only on a sufficiently small set (Although it’s not enough to say “on a set of measure $0$” because the image of a measurable map need not be measurable. Maybe this is difficult). Or you can just demand that every map is a surjection, like Terry Tao does in these notes.) Also the category of random variables, which is the coslice around $\Omega$ in this category.
Peter,
it seems that you created an entry all arrows monic for the sole purpose of making those keywords take the reader to the entry left cancellative category. Is that right?
In this case better to use the redirect functionality provided by the software: adding to the source code of any entry X a line of the form
[!redirects alternative entry name]
has the effect that calling the URL
https://ncatlab.org/nlab/show/alternative entry name
takes the reader to the given entry X
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