# Questions tagged [diagram-chasing]

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

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### Diagram chasing on borsuk ulam theorem proof

I'm trying to understand a proof on the Borsuk Ulam theorem, and it uses the fact that a continuous function from the sphere to the sphere induces a morphism on the Homology long exact sequences, as ...
0 votes
0 answers
49 views

• 5,001
2 votes
0 answers
78 views

### How does a pushout and an exact sequence of modules give another exact sequence?

In this question and this one, it has been asked how an exact sequence of modules $0 \to K \to A \to F \to 0$ and the pushout of $P \leftarrow K \to A$ give rise to a second exact sequence, namely a ...
• 181
3 votes
2 answers
171 views

### Why does the inner square commute if all outer squares commute?

In proving a change of basis theorem in linear algebra, our professor draw this diagram and simply stated that because all the outer squares in this diagram commute, the inner square (green) must ...
• 1,288
2 votes
1 answer
62 views

### If three functions in commutative diagram are bijection, is the fourth one too?

Let's say we have a commutative diagram as in the following picture. The functions $f, h, g$ are all bijections. Can we conclude that $k$ is also a bijection? I need this as part of my proof, but in ...
• 1,288
4 votes
2 answers
291 views

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1 vote
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### About the proof of nine-lemma using snake lemma

I cannot follow the proof $(3\times 3)$ nine-lemma by using the snake lemma. Is there another way to understand the commutativity of the diagram after replacing the morphisms coming from the snake ...
2 votes
2 answers
94 views

• 7,778
0 votes
1 answer
193 views

### any cofibration $i:A \to B$ is a homeomorphism onto its image (question regarding the inverse map)

I was recently working on a problem that introduced the homotopy extension property as a cofibration $i:A \to B$. Let's say we are given the commutative diagram: Now, if $i:A \to B$ is the inclusion ...
• 2,126
-1 votes
1 answer
97 views

### Suppose that $g$ is isomorphism. Then prove that $f$ is monic and $h$ is epic.

$$\\ 0 \to A \to B \to C \to 0 \\ 0 \to A' \to B' \to C' \to 0$$ These are exact and they occur a commutative diagram by homomorphism. $$g=B\to B'\\ f=A\to A'\\ h=C \to C'$$ Suppose that $g$ is ...
• 57
2 votes
1 answer
107 views

### $a \times \mathbf 1 \cong a$ in categories admitting products and having a terminal object $\mathbf 1$

I'm practicing my diagram chasing and reasoning skills, and, as an exercise, I'm trying to prove that if a category has products and also has a terminal object $\mathbf 1$, then for any $a$ an object ...
• 1,553
1 vote
0 answers
60 views

### Category with multiple classes/types/labelled morphisms?

Is there a notion of a category with "labelled" or "typed" morphisms? I imagine that each morphism would belong to a certain class/type, and those classes would form a commutative monoid under ...
3 votes
1 answer
59 views

1 vote
1 answer
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### Making a diagram commute

Suppose we have a $k$-vector space $A$, two linear maps $$T_1, T_2:A\to A$$ and a bilinear map $\mu:A\otimes A\to A$. Is there a way to explicitly construct a map $F$ such that F\circ \mu = \mu\...
• 1,271
1 vote
1 answer
57 views