Timeline for "Monoid objects" without points
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 24, 2017 at 9:16 | comment | added | Simon Henry | @მამუკაჯიბლაძე : My point was just that the inclusion of the category of finite sets or of sets smaller than an inaccessible cardinals are not geometric morphisms because they do not have an adjoint. But they are logical morphisms. I agree that the category of sets is not terminal in the category of logical morphisms. | |
| Dec 24, 2017 at 0:09 | vote | accept | Valery Isaev | ||
| Dec 23, 2017 at 18:58 | comment | added | მამუკა ჯიბლაძე | @SimonHenry Sorry could you elaborate about inclusion functor? I had in mind geometric morphisms, if there is a geometric morphism to a well-pointed topos like finite sets or all sets, it is unique (since every object there is a coproduct of terminals). As for logical functors - is not the trivial topos terminal there? | |
| Dec 23, 2017 at 16:38 | comment | added | Simon Henry | @მამუკაჯიბლაძე : you mean logical morphisms, not geometric morphisms. (those inclusion functor do not have adjoints) | |
| Dec 22, 2017 at 21:35 | comment | added | მამუკა ჯიბლაძე | An example: elementary toposes and geometric morphisms. It has image factorizations, so in particular idempotents split. The topos of finite sets, as well as sets below an inaccessible cardinal, are subterminals. | |
| Dec 22, 2017 at 18:03 | comment | added | Simon Henry | Hum... So what you proved is that in Cauchy complete category those are exactly the monoid objects of slice categories by subterminal objects. Nice ! | |
| Dec 22, 2017 at 17:57 | comment | added | Simon Henry | The empty set is an exemple in the category of sets. If G is a group object in $C/X$ and $X \rightarrow Y $ is a mono then G is a non trivial exemple of the notion in $C/Y$ | |
| Dec 22, 2017 at 17:56 | history | edited | Anton Fetisov | CC BY-SA 3.0 |
added 19 characters in body
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| Dec 22, 2017 at 17:54 | comment | added | Anton Fetisov | @მამუკაჯიბლაძე Are there idempotent-complete examples of such categories? In any case, $T$ will be terminal for the subcategory of objects that admit a map to $X$, which is as good as we can expect. | |
| Dec 22, 2017 at 17:46 | comment | added | მამუკა ჯიბლაძე | There might be $A$ without any maps to $X$, so $T$ is only subterminal. There are categories with lots of subterminals but without a terminal. What is true is that $X$ is a monoid in the slice over $T$ (more or less as in my comment). | |
| Dec 22, 2017 at 17:43 | history | answered | Anton Fetisov | CC BY-SA 3.0 |