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Stream: learning: questions

Topic: Notations for unique morphisms

Ruby Khondaker (she/her) (Mar 28 2026 at 18:28):

Plenty of morphisms in category theory can be described as “the unique one making this diagram commute”. For example, given any pair $f : Z \to A$ and $g : Z \to B$, there exists a unique $u : Z \to A \times B$ such that $\pi_A \circ u = f, \pi_B \circ u = g$.

Sometimes, it’s convenient to then regard $u$ as a _function_ of $f$ and $g$. But for this, it’s necessary to give some _name_ to this function, or at least some notation for it. One example I’ve seen is $(f, g)$, though that gets overloaded with wanting to just consider the ordered pair $(f, g)$. Indeed, with this notation, the function looks like $(f, g) \mapsto (f, g)$ which is indistinguishable from the identity!

Do people have preferred notations for this kind of thing? Personally I like $[f, g]$ since it looks like a list.

I also don’t just mean this for products specifically - I’m interested in finding good names and/or notation for all these “unique morphism” functions:

• For coproducts, I was thinking something like $\text{cases}(f, g)$, since you define the new function by cases in the category of sets.
• For morphisms involving initial/terminal objects, I’m still not sure
• For restricting the codomain of a function to a subset $S$ containing the image (sometimes called corestriction), I think $f |^S$ is standard? More generally you could use this for the unique factorisation of $f$ through a monomorphism if one exists.
• For descending a function respecting an equivalence relation to a map on the quotient, I was thinking something like $f_{/R}$ or $f/q$.
• For limits and colimits, I’m still not sure what the appropriate notation should be.
• For representing objects more generally, I’m still not sure what a good _name_ to give the representation is.

I’d love to hear people’s thoughts and preferred conventions for naming these!

Ralph Sarkis (Mar 28 2026 at 18:52):

I agree with Nathanael's answer here. I also like to call this "unique morphism" the mediating morphism.

Ruby Khondaker (she/her) (Mar 28 2026 at 18:55):

Interesting, so that gives some suggestions for the mediating morphisms for products and coproducts - namely, $\langle f, g \rangle$ and $[f, g]$. Not entirely sure how I feel about the second one since it makes me think more of lists though.

Then I suppose my question can also be phrased as - what is good notation for these “mediating morphisms”?

Ralph Sarkis (Mar 28 2026 at 18:57):

Also, in these particular cases, I (and other people) call them the pairing (product) and copairing (coproduct) of $f$ and $g$.

John Baez (Mar 29 2026 at 20:01):

For products people often call the mediating morphism $(f,g)$ since when $f$ and $g$ are points (emorphisms from $1$) that's what everyone always call it. For coproducts I usually see $\langle f, g \rangle$, so I've adopted that.

David Michael Roberts (Mar 30 2026 at 02:04):

I use the same as John. An alternative for the map out of a coproduct is $[f,g]$, which perhaps coincidentally, perhaps not, matches up with how you would write a linear map $V_1\oplus V_2 \to W$ given linear maps $f\colon V_1\to W$ and $g\colon V_2 \to W$.

At all costs I avoid $(f,g)$ for the map out of the coproduct.

David Michael Roberts (Mar 30 2026 at 02:04):

Also, when writing diagrams, if there is enough scaffolding, I sometimes just decorate the unique map induced by a universal property by $!$

David Michael Roberts (Mar 30 2026 at 02:05):

But for the example of a descended map out of a quotient, for instance, I might try to set things up so that the original map has a tilde, and the descended map doesn't.

David Michael Roberts (Mar 30 2026 at 02:09):

I agree with the corestriction notation $f\big|^T$ for something that's as good as anything, the relation to restriction $f\big|_S$ is pleasing (also, this notation is reminiscent of how people write objects of morphisms in (internal) groupoids with source and/or target coming from some specified set. If you have an internal groupoid $X$ and subobjects $$S,T$$$ of objects, you have $X_S$, $X^T$ and $X_S^T$ as those objects of arrows with source in $S$, with target in $T$, and with source in $S$ and target in $T$, respectively)

David Michael Roberts (Mar 30 2026 at 02:09):

Initial and terminal object I'd just write $0\stackrel{!}{\to} A$ and $A \stackrel{!}{\to} 1$ respectively.