A recent project has forced me to consider a rather special object in a rather nasty category. Consider any category $\mathcal{C}$ which has

  • image objects, meaning for each morphism $f: x \to y$ there exists an object $\text{im }f$ along with morphisms $$x \stackrel{s}{\twoheadrightarrow}\text{im }f \stackrel{i}{\hookrightarrow} y$$ so that $s$ is surjective, $i$ is injective and $f = i\circ s$, plus the obvious universal property,

  • an initial object $\iota$ so that each $x \in \mathcal{C}_0$ admits a unique morphism $\iota \to x$, and

  • a final object $\phi$ so that each $x \in \mathcal{C}_0$ admits a unique morphism $x \to \phi$.

Note that in my case $\iota \neq \phi$, so there is no zero object in $\mathcal{C}$. What we do have instead, is a unique morphism $\iota \to \phi$. This morphism has an associated image object, which I will label $d$.

Has $d$ been defined and studied somewhere? What are its fundamental properties?

I'm sorry if the question is seen as an obvious reference request. Category theory seems particularly rich in this Rumplestiltskin phenomenon, where the difference between finding what you are looking for and flailing around miserably lies simply in knowing a special name.

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    $\begingroup$ When you say "surjective" do you mean "epimorphism" or "regular epimorphism"? How about "injective"? $\endgroup$ Jul 3 '13 at 20:12
  • $\begingroup$ @NeilStrickland, by (in, sur)jective I mean only that the relevant (left, right) cancellation property holds. $\endgroup$ Jul 3 '13 at 20:16
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    $\begingroup$ Which of the two obvious universal properties is the obvious universal property to which you refer? $\endgroup$ Jul 3 '13 at 20:19
  • 3
    $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Jul 3 '13 at 20:34
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    $\begingroup$ 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). $\endgroup$ Jul 3 '13 at 21:01

One context where an object like your $d$ has been studied at least a bit is in quasitopos theory. I doubt this will be very helpful to you since your "nasty" category is probably not a quasitopos, but perhaps it is interesting and may give some ideas.

Unlike in a topos, in a quasitopos $E$ the map from the initial object $0$ to the terminal object $1$ need not be a strong monomorphism (the relevant notion of "subobject" for a quasitopos). If we factor this map as an epimorphism $0\to d$ followed by a strong mono $d\to 1$, then the slice category $E/d$ and the co-slice category $d/E$ are both quasitoposes (these are two of the four subcategories identified by Kevin in the comments), and additionally $E/d$ is a preorder (hence a Heyting prealgebra) while $d/E$ has disjoint coproducts. Moreover, $E$ is the Artin gluing of these two quasitoposes along the functor $F(A) = A^d + d$ from $E/d$ to $d/E$; see A2.6.7 of Sketches of an Elephant.

  • $\begingroup$ This is very cool. In the ~6 years that have elapsed since this question was asked, I have managed to forget what category I was banging my head against at the time. Here's hoping that I find my notes, and that it turns out to be a quasi-topos. $\endgroup$ Mar 22 '19 at 20:00

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