# 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### $a \times \mathbf 1 \cong a$ 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 ...
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### 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 ...
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From Rotman's Algebraic Topology: Consider a commutative diagram with exact rows: $\dots \rightarrow A_n \xrightarrow {i_n} B_n \xrightarrow {p_n} C_n \xrightarrow {d_n} A_{n-1} \rightarrow \... 0 votes 0 answers 73 views ### Doubt on a particular commutative diagram using the Tensor Product construction I've posted two other questions*$$, discussing and asking about the Tensor Product construction, in particular the "canonical construction", which is the one$$: $$\frac{F(V \times W)}{... 2 votes 1 answer 58 views ### Creation of limits and diagram chasing Let T be a monad in X. I want to prove that the forgetful functor G from category of T-algebras on X to X creates limits. I have read that it can be done via "diagram chasing". At this point, I am ... 5 votes 1 answer 213 views ### Naturality of the ker-coker sequence of snake lemma in an abelian category An important part of the snake lemma is the naturality of the ker-coker sequence produced by it. However, no source seems to state or prove this part for arbitrary abelian categories. However, it is ... 2 votes 1 answer 90 views ### How to show that a^1 \simeq a for all C-objects a in a cartesian closed category? (only one direction of isomorphism proof is needed) I have already proven that a \times 1 \simeq 1 \times a \simeq a given a terminal object 1 of a cartesian closed category C (CCC). By CCC I mean that C is finitely complete and has ... 1 vote 1 answer 41 views ### In a topos, an internally functional relation induces a partial morphism As a simple exercise, I wanted to write up a proof that in any topos \mathcal{E}, given a subobject r:R\rightarrowtail X\times Y in \mathcal{E}, if$$(\forall x:X)(\forall y,z:Y)([R(x,y)\wedge R(... 1 vote 1 answer 43 views ### Proving exactness not at the connecting hom in a Snake Lemma subproblem. Here's a picture of the Snake Lemma from nLab: I'm having trouble showing exactness at$\ker g$and haven't even got to the connecting hom yet. So let's focus on$\ker g$. Let$q : A' \to B'$be ... 1 vote 1 answer 58 views ### A question about homotopy pushout Let \begin{array}{ccc} X & \xrightarrow{} &Y \\ \downarrow & & \downarrow \\ Z & \xrightarrow{} & W\end{array} be a commutative diagram in a proper model catgeory and$P$be ... 1 vote 1 answer 39 views ### Injective map between cokers Given a commutative diagram of vector spaces with exact rows I'd like to understand why knowing that$h$is injective proves that the injection$ coker (\alpha) \to coker (\beta)$is equivalent to$...
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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\...