A cartesian diagram? - Mathematics Stack Exchange
A cartesian diagram? Ask Question Asked 8 years, 5 months ago. Modified 8 years, 5 months ago. Viewed 998 ...
abstract algebra - A tower of Cartesian Products is Cartesian ...
2020年2月15日 · I am trying to figure out the following exercise from Vakil's notes: If the two squares in the following commutative diagram are Cartesian diagrams, then the "outside rectangle" (involving U,V,Y, and Z) is also a Cartesian diagram.
Chow's Lemma proof in Milne's Book: Cartesian diagram
2023年9月14日 · First of all I don't understand what is the upper map of the first diagram. Secondly, I'm really not familiar with "cartesian diagrams" and "pullbacks/pushouts". Could you explicitate this reasoning a bit more?
The "magic diagram" is cartesian - Mathematics Stack Exchange
The "magic diagram" is cartesian. Ask Question Asked 10 years, 8 months ago. Modified 7 months ago. Viewed ...
Another proof of the fact that the "magic diagram" is cartesian.
2021年10月4日 · The "magic diagram" has been given one (or more) proof on this site using universal properties of fiber products and "category theory". One would like an "elementary" and "direct" proof of this fact that is student friendly and "non-category theoretic": The "magic diagram" is a Lemma involving fiber products of schemes and one would like a ...
Demystifying the Magic Diagram - Mathematics Stack Exchange
2021年3月12日 · Vakil calls the following pullback diagram the magic diagram. I have also seen it being called the magic square. It often shows up in fiber product diagram chases such as those associated with separatedness assertions. $\require{AMScd}$ \begin{CD} X_1\times_Y X_2 @>>> X_1\times_Z X_2\\ @V V V @VV V\\ Y @>>> Y \times_Z Y \end{CD}
What is the diagram for the limit that is the Cartesian product?
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When is a pullback also a pushout? - Mathematics Stack Exchange
2011年11月12日 · $\begingroup$ 1) Sure, a square that is both a pull-back (cartesian) and a push-out (cocartesian) is called a bicartesian square (Freyd called them push-me pull-you first, then later Doolittle diagrams...). 2) I don't think that there is a general characterization without further assumptions on the morphisms and the surrounding category.
Cokernels in Co-Cartesian Diagram are isomorphic.
We assume the category is abelian. Let the below be a co-cartesian diagram. A co-cartesian diagram is just the dual of a cartesian diagram. \\begin{array}{ccc} A& \\xrightarrow{f} & B ...
solution verification for a proof of "the magic diagram"
2023年2月28日 · There are already many threads about this result, like The "magic diagram" is cartesian, i.e, the following diagram is a pullback: $$ \require{AMScd} \begin{CD} X_1\times_Y X_2 @>{p}>> X_1\times_Z X_2\\ @V{h=f_1\circ y_1=f_2\circ y_2}V V @V{q}V V\\ Y @>{d}>> Y \times_Z Y \end{CD} $$ Using the notation from the following diagram