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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.

6
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1answer
258 views

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 ...
4
votes
1answer
70 views

Diagrams in category theory: formalizing a concept in diagram-chasing

Lemma 1.6.11. Suppose $f_1,...,f_n$ is a composable sequence - a "path" - of morphisms in a category. If the composite $f_kf_{k-1}...f_{i+1}f_i$ equals $g_m...g_1$ for another composable sequence of ...
2
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1answer
59 views

A question about the functoriality of the module of derivations on the category of algebras

Assume that all rings are commutative with identity. Let $k$ be a fixed ring with $k$-algebras $\varphi: k \rightarrow R$ and $\tau: k \rightarrow A$. Let $\tau' : R \rightarrow B$ define an $R$-...
0
votes
1answer
143 views

Using the Snake lemma to prove an Extension

I am trying to prove that $E'$ is an extension of $Q$ by $N'$ \begin{array} 00 &\longrightarrow & N & \overset{i_1} \longrightarrow & E & \overset{\pi_1} \longrightarrow& Q &...
3
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0answers
136 views

Diagram chasing in Abelian categories?

In the nLab page, a technique so-called generalized elements is introduced, which is identical to that on MacLane's Categories for the Working Mathematician. We know that in this method, one can check ...
3
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0answers
172 views

Help to write a proof (category theory diagram)

It is known that $f$, $g$, $h$ are isomorphisms. It is known that $g\circ f = h^{-1}$. I need to write down the proof of the following theorem. I am an amateur mathematician and am not an expert in ...
2
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0answers
108 views

How would one solve Weibel 1.3.1 in a general Abelian category?

Working through Weibel's Introduction to Homological Algebra, I am frequently unsure when it is acceptable to prove results using diagram-chasing and elements, and when Weibel has in mind a category-...
2
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0answers
267 views

How to verify commutativity of a diagram?

Let $C$ be a category, and let all objects $X_i, Y_j$ belong to $ob(C)$, and morphisms $f_{ij}, h_i, g_{ij}$ be morphisms between them in $C$. Let us have a diagram then: How do we verify it's ...
2
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0answers
90 views

Proof that the dual of a coalgebra is an algebra via commutative diagrams

We know that the algebra and coalgebra axioms are given via following commutative diagrams (algebra $A$ and coalgebra $C$ are over a field $\mathbb{K}$): I am now trying to show that the dual of a ...
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0answers
59 views

Graphs and Feynman diagrams: common properties?

I would like to know whether it is possible to establish a clear parallelism between the main features of graph theory (connected and disconnected graphs, graphs with loops, etc) and the 5 basic types ...
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0answers
103 views

Proving that the diagram commutes

Let's consider a continuous map $f: X \rightarrow Y$, so that $Y= U_{1} \cup U_{2}$ -- covering by open sets and $X = f^{-1}(U_{1}) \cup f^{-1}(U_{2}) = V_{1} \cup V_{2}$. How to prove that the ...
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0answers
110 views

commutative diagram with one injective map

Let $A,B,C,D$ be R-modules. Suppose there is a commutative diagram as the following. $A\to B$ is the injective, $A\to C$ and $B\to D$ are surjective. Is $C\to D$ injective?
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0answers
158 views

Extensions and Pushouts using an exact sequence of sets

This might seem a strange way of doing things, that is, inventing a possible example (according to comments, there is no such thing as an exact sequence of sets), but let us try to make one for ...