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Stream: learning: questions

Topic: initial sources as initial objects

Bernd Losert (Apr 23 2023 at 11:42):

In Joy of Cats, they define the concept of a source and the concept of a U-initial source, where U is a "forgetful" functor. I was wondering if this concept can be phrased as a universal mapping property (or as an initial object in some category).

Ralph Sarkis (Apr 23 2023 at 12:15):

It is written a bit after the definition of $G$-initial sources (in 10.58, but I know some copies have different numberings), that when $G = U$ is the forgetful functor for some concrete category, then the $U$-initial sources are the same thing as initial sources defined in 10.41. Then in 10.42, they say that initial 1-sources are the same thing as initial morphisms defined in 8.6. In that definition there is a footnote saying [my editorialization] that the terminology for initial morphisms does not come from an obvious way to see them as initial objects in some category.

Ralph Sarkis (Apr 23 2023 at 12:15):

This does not answer your question but it points towards an easier angle, can we find a way to see initial morphisms as initial objects of some category. Let me reproduce the definition of initial morphism because I don't think it is well-known and it will save other readers a trip to their copy of the book.

Definition: In a category $\mathbf{A}$ concrete over $\mathbf{X}$ with forgetful functor $U$, a morphism $f \in \mathrm{Hom}_{\mathbf{A}}(A,B)$ is an initial morphism provided that for every object $C \in \mathrm{Ob}(\mathbf{A})$ and morphism $g \in \mathrm{Hom}_{\mathbf{X}}(UC,UA)$, if $Uf \circ g$ is in the image of $\mathrm{Hom}_{\mathbf{A}}(C,B)$ under $U$ then $g$ is in the image of $\mathrm{Hom}_{\mathbf{A}}(C,A)$ under $U$.

Ralph Sarkis (Apr 23 2023 at 12:16):

To get more intuition, my favorite examples are in concrete categories over $\mathbf{Set}$:

• In the category of posets, the initial morphisms are functions that are order-embeddings, i.e. $a \leq b \Leftrightarrow f(a) \leq f(b)$ (which forces injectivity, but that is not the important part for initial morphisms).
• In the category of metric spaces and nonexpansive maps, the initial morphisms are isometries (i.e. $d(a,b) = d(f(a),f(b))$ (same remark for injectivity).

Informally, an initial morphism (for concrete categories over $\mathbf{Set}$) is a function $f$ that preserves properties such that when you precompose it with any function $g$, the result preserves properties if and only if $g$ preserves properties.

Mike Shulman (Apr 23 2023 at 17:14):

I think most people call that a [[cartesian morphism]] (for a faithful functor).

Ralph Sarkis (Apr 23 2023 at 17:55):

I guess Lemma 3.1 in that page is a justification for that name, but I think there are typos. I can fix these and add a reference to the Joy of Cats. In the Related Pages, they say cartesian morphisms are precisely initial lifts of a singleton structured cosink. Am I right in saying it should be strictly initial lifts?

Mike Shulman (Apr 23 2023 at 20:15):

I suspect that the original justification of the name is that the cartesian arrows in the [[codomain fibration]] are the pullback squares, a.k.a. "cartesian squares", but I don't know for sure. I wouldn't argue that it's a particularly good name, but it's very standard and other attempts to change it have mostly fizzled.

Mike Shulman (Apr 23 2023 at 20:17):

And with definitions as given at [[final lift]], I believe a cartesian morphism should be a strictly final lift of a singleton sink.

Bernd Losert (Apr 23 2023 at 23:08):

There's a definition of U-initial source that I found in the paper Topological Categories by Brümmer that looks almost like a universal mapping property:

Let $(A, U)$ be concrete. A source $(f_i : X \to X_i)_I$ in $A$ is $U$-initial if for every other source $(g_i : Y \to X_i)_I$ and every $h : UY \to UX$ satisfying $\forall i.\ Uf_i \circ h = Ug_i$, there exists a unique $h' : Y \to X$ such that $Uh' = h$ and $\forall i.\ f_i \circ h' = g_i$.

I have no clue how to consolidate the stuff about $h$ in this definition to make it into a genuine universal mapping property though.

Mike Shulman (Apr 23 2023 at 23:42):

Looks like a universal mapping property to me. What is your definition of "genuine universal mapping property"?

Ralph Sarkis (Apr 24 2023 at 08:11):

Mike Shulman said:

And with definitions as given at [[final lift]], I believe a cartesian morphism should be a strictly final lift of a singleton sink.

I don't think so. Now I think it should be a strictly final lift of a singleton cosink.

Bernd Losert (Apr 24 2023 at 08:20):

If you slightly alter the definition I quoted, you get the universal mapping property of a product:

A source $(f_i : X \to X_i)_I$ is a product if for every other source $(g_i : Y \to X_i)_I$, there exists a unique $h' : Y \to X$ such that $\forall i.\ f_i \circ h' = g_i$.

Ralph Sarkis (Apr 24 2023 at 08:52):

Here is an attempt at making it fit my intuition for genuine universal mapping property. I will do it for $I$ being a singleton for simplicity.

Let $(\mathbf{A},U)$ be concrete and $f: A \to B$ a morphism in $\mathbf{A}$ (also called a singleton source). We define the category of lifts of $f$ as follows.

• Objects are pairs of morphisms $g: UC \to UA$ and $h: C \to B$ such that $Uf \circ g = Uh$. Since $U$ is faithful, we can also say an object is a morphism $g: UC \to UA$ such that there exists $h: C \to B$ satisfying $Uf \circ g = h$.
• Morphisms between $g:UC \to UA$ and $g': UC' \to UA$ are morphisms $m: C \to C'$ such that $h' \circ m = h$ and $g' \circ Um = g$.

Here is a diagram summarizing the situation.
image.png

Now, for any singleton source $f: A \to B$, $\mathrm{id}_{UA} : UA \to UA$ is in the category of lifts because there exists $h = f$ such that $Uf \circ \mathrm{id}_{UA} = Uh$. A singleton source $f: A \to B$ is $U$-initial if and only if $\mathrm{id}_{UA}$ is final in the category of lifts. Let me unroll the two sides to make it clear why they are equivalent:

• $f$ is $U$-initial means for every $g: UC \to UA$ such that there is $h: C \to B$ satisfying $Uh = Uf \circ g$, there exists a unique $\bar{g}: C \to A$ such that $h = f \circ \bar{g}$ and $U\bar{g} = g$ (Definition 10.57 in Joy of Cats).
• $\mathrm{id}_{UA}$ is final in the category of lifts means that for every object in that category, i.e. $g: UC \to UA$ such that there is $h: C \to B$ satisfying $Uh = Uf \circ g$, there exists a unique morphism from $g$ to $\mathrm{id}_{UA}$, that is a unique morphism $\bar{g}: C \to A$ such that $$f \circ \bar{g} = h$ and $\mathrm{id}_{UA} \circ U\bar{g} = g$ or equivalently $U\bar{g} = g$.

Here is a diagram summarizing the situation of the second item:
image.png

Bernd Losert (Apr 24 2023 at 13:31):

@Ralph Sarkis I think that works quite well. I already had the suspicion that being "U-initial" is actually a "final" concept rather than an "initial" concept. Thanks a lot.

Mike Shulman (Apr 24 2023 at 15:29):

Ralph Sarkis said:

Now I think it should be a strictly final lift of a singleton cosink.

Yes, you're right.

Mike Shulman (Apr 24 2023 at 15:30):

Does JoC really call this notion "$U$-initial"?? As you say, it's clearly a "final" concept, not an initial one.

Ralph Sarkis (Apr 24 2023 at 15:33):

That's what I am thinking. In the definition of initial morphism which is what the nlab (and according to you most people) call cartesian morphism, they have this footnote:
image.png
image.png

Ralph Sarkis (Apr 24 2023 at 15:35):

To be honest, the way their definition is stated obfuscates the universality to me.

Mike Shulman (Apr 24 2023 at 16:11):

Ah, I guess their terminology comes from initial topology, which probably predates "initial object".

Mike Shulman (Apr 24 2023 at 16:12):

That's hugely unfortunate.

Reid Barton (Apr 24 2023 at 16:19):

That is very confusing. I wonder how the terminology originated. Maybe it is related to the fact that the ordering of topologies on a fixed set X by inclusion (as subsets of the power set of X) is opposite to the category of topological spaces with underlying set X?

Mike Shulman (Apr 24 2023 at 17:30):

Seems possible. In any case, it would probably be better to avoid the "initial/final" terminology for liftings entirely in the future. What about generalizing the terminology of cartesian arrows and talking about "cartesian cosinks" and "opcartesian sinks"?

Bernd Losert (Apr 25 2023 at 07:37):

Yes, I think this terminology comes from to the usual subset ordering on topologies. The way I rationalize this terminology is like this: "initial" means we are talking about sources and initial objects are like sources since they have arrows coming out of it to all other objects.