# Questions tagged [diagram-chasing]

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

52 questions
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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 ...
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### Vertical and top arrows??

Here in her short note in the middle of the page 1, Emily Riehl says ...with the top two vertical arrows ... I just wonder how vertical arrows can be top or bottom. I would say that vertical arrows ...
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### Why is this universal map in a proof of the co-Yoneda lemma actually natural?

I'm attempting to prove that every presheaf is a canonical colimit of representable presheaves by constructing a limiting cocone directly (I'm aware that there are more elegant proofs, but this is ...
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### Category theorists: would you use a software tool for diagramming / chasing?

Here is a video demo of my work so far. It's written in PyQt5 and consists of 30 files so far. I have LaTeX support written already but you don't see that in this demo. Was wondering if you find ...
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### 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-...
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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, ...
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### The Existence of a Natural Isomorphism Confirming the Existence of a Limit, as a Commutativity Diagram

I am reading Category Theory for Programmers and I am having some trouble in Part 2, Chapter 2: Limits and Colimits. The author writes: Now that we have two functors, we can talk about natural ...
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### 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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### 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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### Interpreting a nonstandard definition of a tree

Definition: A tree is a triple $(T,\sigma,\pi)$ where $T$ is a set and $\sigma$ is a so-called successor function from $T$ to the set $T^*$ of all nonempty subsets of $T$, together with a surjective ...
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### 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 &...
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### 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 ...
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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 ...
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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 ...
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### Does zero-kernel imply monic in Abelian categories?

I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ...
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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'} @>>{...
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### A functor on commutative diagrams

As suggested by Daniel Rust I'll pose this as a separate question. Let $C$ be a category. Denote by $Ar(C)$ the following category: an object in $Ar(C)$ is a morphism $X_1 \rightarrow X_2$ in $C$. ...
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### Diagram chasing, and more

1) Assume that $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \rightarrow 0$ and $0 \rightarrow C_1 \rightarrow C_2 \rightarrow D \rightarrow 0$ are exact, $i=1,2$. Show, using a diagram chase, ...
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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 ...