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Stream: learning: questions

Topic: Pushouts and Products

Julius Hamilton (Dec 05 2024 at 20:36):

I am just trying to synthesize what I was just reading about, and open to corrections.

For 2 objects $X, Y$, their product is an object $X \times Y$ and 2 morphisms $\pi_X : X \times Y \to X, \pi_Y : X \times Y \to Y$, such that for any object $Z$ and morphisms $f : Z \to X, g : Z \to Y$, there exists a unique morphism $i : Z \to X \times Y$ such that $\pi_x \circ i = f$ and $\pi_y \circ i = g$.

When a morphism $f : X \to Y$ is equal to a composition of morphisms $(g : Z \to Y) \circ (h : X \to Z)$, I think one can say $f$ factors into $g \circ h$, and also $f$ factors through $Z$.

If two objects have a product $X \times Y$, then for any object $Z$ and pair of morphisms $f : Z \to X, g : Z \to Y$, $f$ and $g$ factor through $X \times Y$.

Three objects $X, Y, Z$ and two morphisms $f : X \to Y$, $g : X \to Z$ is called a span.

The product $X \times Y$ with projection morphisms $\pi_X, \pi_Y$ forms a span.

For a span $f: X \to Y, g : X \to Z$, if there were a term for an object $W$ and morphisms $d : Y \to W, e : Z \to W$ such that $d \circ f = e \circ g$, I think one may be able to define the pushout as the one of those which all the others factor through (if it exists).

How do products/coproducts relate to spans/cospans and pullbacks/pushouts?

A limit is a universal cone.

A cone over a diagram $D$ is an object $C$ with a morphism $c_i$ to each object in the diagram $D$, I think such that all paths commute.

I think one may define universal to mean, for a collection of diagrams with the same structure (for example, cones), the diagram that every diagram factors through is the universal one.

I think that the product $X \times Y$ is the limit of spans $f : W \to X, g : W \to Y$.

Could it be that the pushout of a span is the colimit?

I expect to have made errors in the above.

David Egolf (Dec 05 2024 at 21:14):

Yes, a pushout is a colimit. Any pushout is a colimit of a diagram that has the "span" shape you're describing. You may find section "3. Definition" of the nLab article I linked to be of interest.

Jencel Panic (Dec 09 2024 at 06:19):

You are mixing the things a bit:

Limits are constructed from diagrams, using cones.

The limit of the diagram 2 is a product.
The colimit of the diagram 2 is a coproduct.

The limit of a cospan is a pullback.
The colimit of a span is a pushout.

And last but not least:
The limit of an empty diagram is the initial object.
The colimit of the empty diagram is the terminal object.

What other limits are there?

Mike Shulman (Dec 09 2024 at 06:21):

(... or an initial/terminal object, as with the other kinds of limits and colimits)

Peva Blanchard (Dec 09 2024 at 07:20):

Another well-known example of a limit is the equalizer of two parallel arrows.
Of course, you have the dual version: coequalizer.