Quick further question about the notion of "pullback/pushout".

Question

Previously, I asked a question about the notion of "pullback" in category theory here, I just have one further quick question about the concept of "pullback/pushout", that is, does the concept always make reference to square shape commutative diagrams, or can it be some other commutative diagram shapes. Thank you in advance.

Answer 1

Pullback generally refers to the square-shaped diagrams.

But! Pullbacks are limits of diagrams that look like this: $$\bullet\longrightarrow\bullet\longleftarrow\bullet$$

Limits of diagrams that look like this: $$\bullet\\\downarrow\\\bullet\longrightarrow\bullet\longleftarrow\bullet$$

Could reasonably also be called pullbacks. More generally, limits of diagrams of the form: one object $X$, some set $I$, for each $i\in I$ an object $A_i$ and a single arrow $f_i:A_i\to X$, could be called pullbacks. However, such a limit could be encoded as a limit of a square diagram of form: $$X\overset{\Delta}{\longrightarrow}\prod_{i\in I}X\overset{\prod_if_i}{\longleftarrow}\prod_{i\in I}A_i$$

Which is a pullback of the usual type, so there is no real difference. It may be a good exercise to check this for yourself.

Pullbacks of the more general form are relevant e.g. in the discussion and use of "intersections" of subobjects. This dualises to unions, which are (generalised) pushouts of the dual diagrams to those above, for instance colimits of something like this: $$\bullet\\\uparrow\\\bullet\longleftarrow\bullet\longrightarrow\bullet$$(Where each arrow is monic, in the specific context of "unions").

Answer 2

Yes, you can have wide pullbacks or pushouts.

Wide pullbacks have more than two 'legs'. See the NLab article to see it described more fully. The wide pushout is its dual.

You also have homotopy pullbacks and pushouts. These make sense in homotopical situations where the ordinary pullback or pushout doesn't make sense. There is an extensive theory about this.