Timeline for What is the image of the intial object inside the final object called?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 22, 2019 at 20:00 | vote | accept | Vidit Nanda | ||
| Mar 21, 2019 at 23:30 | answer | added | Mike Shulman | timeline score: 3 | |
| Jul 3, 2013 at 21:01 | comment | added | Kevin Ventullo | So based on the four possibilities, you can decompose $Ob(\mathcal{C})$ as a disjoint union of four groups, arranged in a 2x2 configuration, so that all morphisms are going from left to right, or top to bottom (or staying inside the group). | |
| Jul 3, 2013 at 20:49 | comment | added | Kevin Ventullo | For every object $X\in\mathcal{C}$, $Hom(X,d)$ is either a singleton or empty, and the same is true for $Hom(d,X)$. | |
| Jul 3, 2013 at 20:34 | comment | added | Todd Trimble | A more usual definition of $im(f)$ is the smallest subobject through which $f: A \to B$ factors. If the category has equalizers, it may be shown that the map from $A$ to (the domain of) $im(f)$ is an epimorphism. | |
| Jul 3, 2013 at 20:22 | comment | added | Vidit Nanda | @StevenLandsburg sorry, I should have specified: given any object $e$ with surjective $x \to e$ and injective $e \to y$ whose composition equals $f$, there exists a morphism $\text{im }f \to e$ (not the other way around) making things commute. | |
| Jul 3, 2013 at 20:19 | comment | added | Steven Landsburg | Which of the two obvious universal properties is the obvious universal property to which you refer? | |
| Jul 3, 2013 at 20:16 | comment | added | Vidit Nanda | @NeilStrickland, by (in, sur)jective I mean only that the relevant (left, right) cancellation property holds. | |
| Jul 3, 2013 at 20:12 | comment | added | Neil Strickland | When you say "surjective" do you mean "epimorphism" or "regular epimorphism"? How about "injective"? | |
| Jul 3, 2013 at 19:04 | history | asked | Vidit Nanda | CC BY-SA 3.0 |