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Questions tagged [diagram-chasing]

For questions about proofs using equivalent map compositions in commutative diagrams in homological algebra, or in category theory in general.

Diagram chasing is a method for proving statements using equivalent map compositions in commutative diagrams, thereby using properties of the diagram such as injective or surjective maps, or exact sequences. Mostly used in homological algebra and category theory. Examples include the five lemma, the snake lemma and the nine lemma.

See also Diagram Chasing at MathWorld.

172 questions
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Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal properties. It was an interest little shared by…
Jesko Hüttenhain
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Looking for student's guide to diagram chasing

I'm teaching myself some category theory, and I find that I'm very slow with diagram chasing. It takes me some times a very long time to decide whether adding an arrow to a diagram preserves the diagram's commutativity, or that a given arrow…
kjo
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Short exact sequence of exact chain complexes

If $0 \rightarrow A_{\bullet} \rightarrow B_{\bullet} \rightarrow C_{\bullet} \rightarrow 0$ is a short exact sequence of chain complexes (of R-modules), then, whenever two of the three complexes $A_{\bullet}$,$B_{\bullet}$,$C_{\bullet}$ are exact,…
Felix Hoffmann
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Is there a computer program that does diagram chases?

After having done many tedious, robotic proofs that various arrows in a given diagram have certain properties, I've begun to wonder whether or not someone has made this automatic. You should be able to tell a program where the objects and arrows…
Helmut
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Why pasting a finite number of commutative diagrams is commutative

I am aware that if you paste two commutative squares, that diagram is commutative, but, in general, how can one prove that a diagram (with squares or triangles) is commutative iff every subdiagram is commutative? I can't see how to generalize.…
HeMan
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Equalizers by pullbacks and products

I'm trying to solve exercise 5.6 in Steve Awodey's "Category Theory": Show that a category with pull-backs and products has equalizers as follows: given arrows $f, g: A \to B$, take the pullback indicated below, where $\Delta = \langle 1_B, 1_B…
Anne
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Fake diagram lemmas that look like they should work but don't.

I got a book on homological algebra in a textbook giveaway and I'm just starting to learn more about exact sequences in preparation for reading the book more seriously. I have seen things like the following before but don't understand them: The…
Greg Nisbet
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Category theorists: would you use a software tool for diagramming / chasing?

Update: Source Code Repository Screenshots: Now the users can edit the default colors of arrows, nodes etc. using the ColorEditor: Users can draw diagrams and store them into a Graph Database (Neo4j) seamlessly. Here are the videos of the old…
Daniel Donnelly
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References to supplement Goldblatt's "Topoi"

I've been using the book "Topoi: A Categorial Analysis of Logic" by Robert Goldblatt. I found it intuitive and fun, but I feel like some parts are missing. For instance, I struggled to prove $a^1 \cong a$, and it seems like I wasn't the only one. In…
Larry B.
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Practical approaches to working with nonplanar commutative diagrams?

The 4-associahedron is the 4-dimensional version of Mac Lane's pentagon diagram. If you look at Trimble's notes on tetracategories, you can see the obvious difficulty in working with such a diagram (using it to verify things like coherence etc). …
user126
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Proving the (Strong) Four Lemma using the Snake Lemma

$\DeclareMathOperator{\im}{im} \DeclareMathOperator{\coker}{coker} \require{AMScd}$ The usual formulation of the Strong Four Lemma is: given the diagram below, if the rows are exact, $\alpha$ is epic, and $\delta$ is monic, then $g(\ker \beta) =…
Henry Swanson
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Category Theory -- limits commute with kernels

This is Exercise 1.6.I from Vakil's notes on Algebraic Geometry. Suppose $\mathscr{C}$ is an abelian category and $a: \mathscr{I}\rightarrow \mathscr{C}, b: \mathscr{I}\rightarrow \mathscr{C}$ are two diagrams in $\mathscr{C}$ indexed by…
KittyL
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What comes after diagram chasing?

An early edition of Lang's algebra textbook gives the famous exercise to Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book. Here are some of theorems you would find in such a book:…
user134824
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Category theory: Enough that polygonal diagrams commute

I've read somewhere that for a categorical diagram to commute, it is enough that all its polygonal subdiagrams commute. I want a reference and a detailed proof of this. Please also give a formal definition of polygonal subdiagrams.
porton
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Help with diagram chasing

Given the diagram $\require{AMScd}$ \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V\alpha VV \#@V\beta V V\# @VV\gamma V @. \\ 0 @>>> {A'} @>>{f'}> {B'} @>>{g'}> {C'} @>>> 0\\ \end{CD} where $\alpha,\gamma$ are monic and the rows are exact.…
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