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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.

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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 ...
24
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3answers
2k views

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 ...
22
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4answers
2k views

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_{\...
20
votes
1answer
626 views

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 ...
10
votes
1answer
334 views

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 (...
8
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2answers
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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 ...
7
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1answer
240 views

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'} @>>{...
6
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1answer
257 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 ...
5
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1answer
292 views

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 ...
5
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1answer
300 views

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 ...
5
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2answers
835 views

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 ...
5
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1answer
69 views

applications of (topological and algebraic) commutative diagrams in organic synthesis

In algebraic topology, there are a lot of commutative diagrams and commutative diagrams up to homotopy. Different ways of compositions of maps in a commutative diagram are equal or homotopy equivalent....
4
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3answers
692 views

How to do diagram chasing effectively?

I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3. I have spent a lot of time ...
4
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1answer
235 views

Step in five-lemma's proof.

Consider the following commutative diagram, where rows are exact: $$\require{AMScd} \begin{CD} M_1 @>f_1>> M_2 @>f_2>> M_3 @>f_3>> M_4 @>f_4>> M_5 \\ @Vh_1VV @...
4
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1answer
93 views

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 ...
4
votes
2answers
61 views

Equaliser of homomorphism of integers

I have the following situation: $ \mathbb{Z} \mathop{\rightrightarrows^{0}_{2} \mathbb{Z}} \ \ $ Where here $0$ and $2$ stands for the multiplication. Now, I want to find the equaliser of this ...
4
votes
1answer
540 views

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 ...
4
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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 ...
3
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3answers
202 views

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, ...
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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1answer
15 views

Can we deduce that $M_0$ is a submodule of the limit of the following diagram?

Let $M_0$ be an R-module, and suppose $M_{n+1}$ is the pushout of the diagram below as shown, for all $n \in \mathbb{N}$: $$\begin{array}{ccc}M_n&\to& M_{n+1}\\\uparrow &&\uparrow\\A&...
2
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1answer
35 views

Can composition of morphisms in a category be carried out on any subgraph of a commutative diagram, in-place?

Here is what the rule looks like to us and how we specify it to the app I'm writing. I was wondering can you take any commutative diagram $J$ and apply this rule to a subgraph matching $A \...
2
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1answer
172 views

Adjunctions and isomorphism.

I'm stuck with the next exercise. I don't know how can I solve it. Let $\mathcal{F}:\mathcal{C}\to\mathcal{D}$ and $\mathcal{E},\mathcal{G}:\mathcal{D}\to\mathcal{C}$ three adjoint functors $\...
2
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1answer
111 views

Triple Products are Isomorphic

I am currently working through Awodey's Introduction to Category Theory, and I'm learning how to move around complicated diagrams. I want to show that $A\times(B\times C)\cong(A\times B)\times C$; ...
2
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1answer
79 views

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 $C$. ...
2
votes
1answer
55 views

Condition for pullback to split

This is probably an elementary question, but I'm new to this machinery. Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $\Gamma$ be another group. Suppose we have a homomorphism $\phi:\...
2
votes
1answer
57 views

Snake lemma for $R$-modules. Help with $\ker$ maps, not connecting hom.

In Construction of the maps they say that the construction of the kernel/cokernel maps comes from the commutativity of the diagram: $$ \require{AMScd} \begin{CD} & & A @>{f}>> B @>...
2
votes
1answer
58 views

Proving a diagram chase result from standard lemmas

Consider three exact sequences of modules over a commutative ring $0\to A\to M\to N\to B\to 0$, $0\to E\to M\to K\to F\to 0$, $0\to C\to N\to K\to D\to 0$, where the maps $M\to N$ and $N\to K$ and ...
2
votes
1answer
89 views

Reference for homological cross lemma?

A colleague and I have recently found ourselves using the following result: $\require{AMScd} \begin{CD} @. @. 0 @. @. \\ @. @. @VVV @. @.\\ @. @. A @. @. \\ @. @. @VV\phi_1 V @. @. \\ 0 @>>> ...
2
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1answer
304 views

Fiber product with diagonal morphism [duplicate]

Stacks tag 01KR states that the diagram of schemes is "by general category theory" "a fibre product diagram". I tried to show this using the universal property, but didn't obtain anything useful. How ...
2
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2answers
154 views

Prove that there's a unique morphism that completes the commutative diagram

I have to prove that there's a unique $\gamma : M'' \rightarrow N''$ that completes this diagram considering the rows are exact. $$\begin{array} MM' \stackrel{f_1}{\longrightarrow} & M & \...
2
votes
1answer
44 views

Commutative hexagonal diagram of Abelian groups; proving a certain equality

I'm trying to prove the following lemma by diagram chasing, but I've had no success, so I decided to ask for help here. Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be Abelian groups, and let $a_{1}$, ...
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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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 $R$-...
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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 ...
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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 ...
1
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1answer
137 views

Cones of a morphism of distinguished triangles

In a derived category (actually, I am interested in the derived category of abelian groups, if that helps), suppose we are given a morphism of distinguished triangles $$\require{AMScd} \begin{CD} X @&...
1
vote
1answer
147 views

In any commutative diagram with isomorphism $f$ we may replace $f$ by $f^{-1}$?

Let $D$ be a diagram in category $\mathcal{C}$ that involves the isomorphism $f$. I don't know how to express that in formal terms. Under what conditions can we replace $f$ in the diagram by $f^{-1}$...
1
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1answer
29 views

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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1answer
65 views

Is the induced map in the following diagram a regular epimorphism?

Consider the following commutative diagram in a semiabelian category where $f,f',\alpha$ and $\beta$ are regular epimorphisms. Is it true that the induced dotted map is a regular epimorphism too? If ...
1
vote
1answer
56 views

Isomorphisms Between some terms of a Long Exact Sequence

Suppose we have two long exact sequences of finite dimensional $k$-vector spaces: $$ 0 \to A_1 \to A_2 \to A_3 \to \cdots $$ $$ 0 \to B_1 \to B_2 \to B_3 \to \cdots $$ And assume that $A_6 \...
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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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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 ...
0
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2answers
92 views

Checking commutativity of a diagram of modules over some ring and what the commutativity of the diagram implies.

Suppose that you have the following diagram of modules over some ring: These are my questions: (1) To prove that the diagram is commutative, we needs to prove that $gf=kh$, $wf=rv$, $zh=uv$, $sw=xg$,...
0
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1answer
62 views

Adjoint functors, inclusion functor, reflective subcategory

Suppose that a category $A$ is reflective in a category $B$ and that the inclusion functor $K:A\to B$ has a left adjoint $F:B\to A.$ Now what does it technically mean that this bijection of sets $$A(...
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1answer
91 views

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 ...
0
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1answer
124 views

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, ...
0
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0answers
109 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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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 &...