# Greatest common factor in a category

I'm looking for a name (or references or search terms) for the following construction.

In a category $\mathcal C$, let a doodle mean a pair $\langle D,\Omega\rangle$ where $\Omega$ is an object and $D$ is a set of morphisms with codomain $\Omega$ (but potentially different domains).

Let a whatsit over a doodle $\langle D,\Omega\rangle$ mean a triple $\langle(\varphi_d)_{d\in D}, A, \psi\rangle$ where $A$ is an object of $\mathcal C$, $\varphi_d$ is a $D$-indexed family of morphisms with codomain $A$, and $\psi$ is a morphism $A\to\Omega$, such that $\varphi_d\psi = d$ for each $d\in D$.

Given a doodle, the whatsits over that doodle are the objects of a category where a morphism from $\langle (\varphi)_d,A,\psi\rangle$ to $\langle (\varphi'_d),A',\psi'\rangle$ is a $\mathcal C$-morphsim $\beta:A\to A'$ such that $\varphi_d\beta=\varphi'_d$ for all $d$ and $\psi=\beta\psi'$.

An initial object of the category of whatsits is a greatest common factor of the doodle. Intuitively, the $\psi$ part of the initial whatsit is the "longest" arrow that is a right factor of every morphism in the doodle.

"Doodle" and "whatsit" are (obviously) my own temporary names.

Question: Do these things have names already? Is it possible to describe them within the framework Wikipedia presents for a universal property?

It would be possible to say that a doodle is simply a whatsit whose third component is an identity morphism (modulo some new way to index the first component of each whatsit), but it not clear to me whether that is useful.

• @goblin: In this analogy a pullback would be a "smallest common multiple" -- the arrows we start with are all factors of the pullback. I need the converse: something that is a factor of each of the initially given arrows. – Henning Makholm Jan 24 '15 at 3:27
• @goblin: Hmm... I think that whatsits may be objects in a slice (not coslice) category over the category of cocones of a discrete diagram. – Henning Makholm Jan 24 '15 at 3:34
• Yep, I think I get you. Given a pair of arrows $f,g : A,B \rightarrow \Omega$ with common codomain, I think you're interested in the coproduct of $f$ and $g$ viewed as objects of the slice category of $\Omega$. When $f$ and $g$ are injective functions, this is the "union" of $A$ and $B$. – goblin Jan 24 '15 at 3:34
• @goblin: Hm, yes, that sounds about right, and less convoluted than my attempt to wrap my head around it. I'd be inclined to accept that as an answer. – Henning Makholm Jan 24 '15 at 3:43
• Okay, I've posted my understanding of the construction. Perhaps something better will come along though. I would Google "colimits in slice categories" in the search for further information. – goblin Jan 24 '15 at 3:50

Fix a category $\mathbf{C}$ and an object $\Omega$ in this category. A $\mathbf{J}$-shaped doodle to $\Omega$ is just a $\mathbf{J}$-shaped diagram in the coslice category $\mathbf{C} \downarrow \Omega$. A whatsit to a doodle $D$ is a cocone to $D$; hence the "greatest common factor" of $D$, being the initial whatsit to $D$, could also be described as the colimit of $D$.
• Okay, so what I'm looking for is simply the coproduct of all the arrows in $D$, viewed as objects of $\mathcal C\downarrow \Omega$. That's much simpler than I could have come up with myself. Thanks, – Henning Makholm Jan 24 '15 at 12:36