Questions tagged [diagram-chasing]
For questions about proofs using equivalent map compositions in commutative diagrams in homological algebra, or in category theory in general.
164
questions
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Commutativity of a diagram involving braids
I have this diagram and I want to prove its commutativity
Let me explain what it means.
$\beta$ is a braid with $n$ strings, that is represented in $\mathbb{D} \times [0,1]$. Its uper ends are $\beta(...
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Mac Lane Chapter 7 Section 2 Exercise 1
Let $\mathcal{C}$ be a monodical category, with the monodical product written $\otimes$, the associator denoted $\alpha$, and the left/right unitors denoted $\iota^\ell,\iota^r$ respectively. Mac ...
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20
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Chasing diagrams of chain complexes which are pointwise projective
For context this question arose in a simple K-theory computation, where I wish to show that the inclusion of the category of bounded degreewise projective chain complexes into the category of bounded ...
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33
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Determining initial algebras and final coalgebras for a given functor without using limits/colimits
I'm trying to find the final coalgebra for a certain functor but I have no idea how to do that in general, so I was hoping to go through the process with some simpler examples. In section 4.1, The ...
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Elegant Proof of Snake Lemma
I'm considering the following diagram
\begin{array}{ccccccccc}
&&&&0&&0&&\\
&&&&\downarrow& &\downarrow& &\\
& && & \...
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Doubt in the Magic Square
So, suppose I work in a category where fibred products exist. Let $X_1,X_2, Y$ and $Z$ objects in this category, $f_i:X_i\rightarrow Y$, and $g:Y\rightarrow Z$ be morphisms.
I am trying to show that ...
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Fake diagram lemmas that look like they should work but don't.
I got a book on homological algebra in a textbook giveaway and I'm just starting to learn more about exact sequences in preparation for reading the book more seriously. I have seen things like the ...
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Comparing order of $\ker f$ and $\ker\tilde{f}$
Let $A$,$B$ be abelian groups. Let $f: A\to B$ be group homomorphism.
Let $C\subset A$ and $D\subset B$ be subgroups of $A$ and $B$.
Let $C\subset D$.
There is a canonical map $\tilde{f}: A/C \to B/D,...
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1
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83
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A question on commutative diagram of abelian groups [duplicate]
I am trying to solve the following problem.
Question: Consider the commutative diagram of abelian groups given below, where the rows in the diagram are exact.
Prove or disprove: if $p,q,s$ and $t$ ...
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A question on diagram of groups [closed]
Question:
Assume that the rows of the following diagram of abelian groups are exact.
Prove or disprove the following statements:
If $p,q,s,t$ are zero homomorphisms, so is $r.$
If $p,q,s,t$ are ...
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How does the following commutative diagram within the following proof match the definition for universal property of the kernel and also what is $g?$
$\color{Green}{Background:}$
$\textbf{Definition:}$ (Universal property of the kernel) Let $R$ be a commutative ring and $f:A\to B$ a morphism of $R-$modules. Recall that the $\textit{kernel}$ of $f$ ...
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Proving that the functor $\mathscr{B} \rightarrow 1$ is limit-preserving. Need help constructing the diagram.
I am newbie to Category Theory and I just read about limit preserving functors and I was trying to prove that the functor $\mathscr{B} \rightarrow 1$ preserves limits. The book I am following is ...
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Follow up question on diagram-chasing/short exact sequences.
Assume I have the following commutative diagram of abelian groups, which is exact.
$\require{AMScd}$
\begin{CD}
A @>\psi>> B @>\varphi>>C @>\pi>> D \\
@V \alpha V V @VV \...
- 1,551
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0
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95
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Commutative diagrams of exact rows.
Assume I have the following commutative diagram of abelian groups, which is exact.
$\require{AMScd}$
\begin{CD}
A @>\psi>> B @>\varphi>>C @>\pi>> D \\
@V \alpha V V @VV \...
- 1,551
6
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2
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191
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What is the most diagrammatic proof that $gf = 0 \implies \text{im } f \subset \ker g$?
The proof of the following fact is trivial to most Arrow theorists / Linear algebraists, but I'm developing software that needs to "understand" in a sense this basic fact, because it is ...
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What are colored morphisms/arrows intended to mean in these diagrams?
I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include ...
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Not understanding concretely the Wikipedia or text definition of "Subobject classifier"
I have two major questions.
In the following diagrams, only some of the arrows make sense to me or are explained. Others I have never seen and are not brought up in any of the webpages or books that I ...
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86
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Extension of a homomorphism from $(H \leq G) \xrightarrow{\varphi} X$ to $G \xrightarrow{\psi}X$ -- Is this a correct diagram?
I want to make sure that I am understanding correctly the idea of a homomorphism $(H \leq G) \xrightarrow{\quad \varphi \quad} X$ being "extended" to a homomorphism $G \xrightarrow{\quad \...
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Need help drawing these diagrams in category theory (sources for good diagrams would also be appreciated)
I already feel that I can better understand mathematical structures by using diagrams from category theory. The books I am reading only give me the basics and I am interested in learning how to draw ...
- 1,080
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1
answer
84
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Commutative diagram between modules and their double dual
Hungerford Algebra, Exercise IV.$4.9$:
For any homomorphism $\phi: A \rightarrow B$ of left $R$-modules the diagram is commutative, where $\theta_A(f)(a)=f(a)$ for $f \in A^*$ and $a \in A$ and $\...
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What can be said about this map in a commutative diagram? [closed]
Suppose that I have the following commutative diagram of maps between sets:
where $u,v$ are bijections. What can be said about the map $\varphi$ in the middle? Is it possible to conclude $\varphi$ ...
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92
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Interpreting Diagrams About Power Object Functor
I'm reading Sheaves In Geometry And Logic and this diagram confuses me:
First, do those diagrams really live in $\mathcal{E}$ resp. $\mathcal{E}^\text{op}$ or rather in $[J^\text{op},\mathcal{E}]$ ...
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1
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186
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Intuition behind large diagrams in category theory
I am attempting to read "Tensor Categories" by Pavel Etingof et. al.
The following pentagon axiom is a part of the definition of a monoidal category:
The following diagram is part of the ...
- 153
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1
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66
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Show that $\phi$ is surjective in commutative staircase diagram
I'm having some trouble with the following problem involving a sort of staircase diagram and I would really appreciate some help.
Assume the following commutative diagram of modules and module ...
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1
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1
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94
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Possibly easy diagram chase of the $3 \times 3$ lemma
Given the following diagram
which is exact on all horizontal rows and the left and middle columns. I'm trying to show that it is exact at $A''$ which amount to showing that $\ker(\varphi_3) = 0$ that ...
- 193
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votes
1
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Theorem 13.1 of Sack's Saturated Model Theory: error in proof?
Theorem 13.1 of Sacks' Saturated Model Theory (1st ed.) says in part, that if every diagram of this type:
can be completed as shown, then $T$ is substructure complete (i.e., $T\cup\text{diag}(A)$ is ...
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How to show the commutativity of diagrams involving homology in abelian categories?
In MacLane's book, it is shown that diagram chases can be made in any abelian category using "members" instead of elements (page 204-208). But I have two problems (concerns) about the ...
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In an abelian category, show that $f(A) = f(B)$ implies $A \subseteq ((A+B) \cap \text{Ker} f) + B$.
Title: In an abelian category, show that $f(A) = f(B)$ implies $A \subseteq ((A+B) \cap \text{Ker} f) + B$.
I'm trying to show this result with as little machinery as possible. This includes not using ...
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68
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Eigenvalues in commutative diagram
Let $K$ be a field.
Suppose, we have linear maps $f: K^n \mapsto K^n$, $g: K^m \mapsto K^m$ and $h: K^n \mapsto K^m$ such that $h$ is surjective and $h \circ f = g\circ h$. Then every eigenvalue of $g$...
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Categorical argument for the exactness of a diagram
$\DeclareMathOperator{\id}{id}$Let $C$ be an abelian category and suppose we have the following diagram in $C$ $\require{AMScd}$
\begin{CD}
@. 0 @. 0 @. 0 \\
@. @VVV @VVV @VVV \\
0 @>>> X_0'@&...
- 1,048
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0
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When is the induced map in a pullback diagram a fibration?
Let $X$, $Y$ and $Z$ be topological spaces and let $f:X \to Z$ and $g:Y \to Z$ be continuous. Let $P$ be the pullback of $X\stackrel{f}{\to}Z\stackrel{g}{\leftarrow} Y$. Let $E$ be another space and ...
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Lifting property for Hausdorff spaces
I was scrolling nlab instead of studying my topology final, and stumbled on the following page: https://ncatlab.org/nlab/show/separation+axioms+in+terms+of+lifting+properties#hausdorff_spaces_
And ...
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3
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2
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133
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Exact sequence induced by filtration
Let’s suppose that $M_1, M_2, M_3$ are $A$-modules with
$$
0 \to M_1 \to M_2 \to M_3 \to 0
$$
exact.
Given $I \subset A$, is true that
$$
0\to M_1 / I^n M_2 \cap M_1 \to M_2 / I^n M_2 \to M_3 / I^...
- 5,174
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0
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89
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How to show excisive triad symmetric from the Eilenberg-Steenrod axioms
Definition of excisive triad: $(X;X_1,X_2)$ is an excisive triad if the inclusion $(X_1,X_1\cap X_2)\rightarrow (X,X_2)$ induces isomorphism in relative homology.
The definition is not symmetric in $...
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3
votes
1
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144
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Difficulty proving the second isomorphism theorem: $\frac{X}{X\cap Y}\cong\frac{X\cup Y}{Y}$ in an Abelian category
$\newcommand{\cok}{\operatorname{cok}}\newcommand{\ker}{\operatorname{ker}}$This is an exercise from Freyd's introduction to Abelian categories here on page $59$ (w.r.t the manuscript). Everything ...
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How do you express that two commutative diagrams can be glued together along a common subgraph and retain commutativity?
I'm wondering about this, because I want to write software that lets you operate on CD's on the computer screen.
I was wondering what existing mathematical tools are required to describe the titled ...
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69
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Does the Barr-embedding preserve coequalizers
Let $\mathbb C$ be a small regular category. Let $J$ be the Grothendieck topology generated by coverings $\{U'\twoheadrightarrow U\}$ consisting of precisely the regular epimorphisms in $\mathbb C$. ...
- 4,350
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49
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Question on the exactness at the third kernel $\ker\nu$ in the Snake Lemma
The commutative diagram is as follow:
$\newcommand{\coker}{\operatorname{coker}}
\newcommand{\im}{\operatorname{im}}
0\stackrel{}{\longrightarrow} \ker\lambda\stackrel{}{\longrightarrow} \ker\mu\...
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What on Earth does Lang mean by "write the second square in the form..." in proof of Lemma 5.2, Homotopies of Morphiphsms of Complexes?
On the top of page 789 (I own the hardcopy of the book btw ;) it says:
Next we must construct $f_1$. We write the second square in the form
$$
\require{AMScd}
\begin{CD}
0 @>>> E^0/M @>&...
- 21.8k
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0
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49
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Why does this diagram commute by uniqueness of the maps?
I do not understand the following remark from the lecture:
Consider two field $L$ and $K$. We know that there exist unique(!) ring homomorphisms $f_K: \mathbb{Z} \rightarrow K$ and $f_L: \mathbb{Z} \...
- 4,212
2
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1
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180
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Short exact sequence of modules induces short exact sequence of cokernels
I have a commutative unital ring $A$, a short exact sequence $L\to M\to N$ of $A$-modules, a faithful exact contravariant functor $D:\text{Mod}_A\to\text{Mod}_A$ satisfying $$DM=0\iff M=0,$$ and ...
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1
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104
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Basic example in diagram chasing
Consider the following diagram:
Given that $h \circ f = k$ and $l \circ h = g$, prove that $ g \circ f = l \circ k $
Answer:
page-35, An introduction to Category Theory, Harold Simmons
I am a bit ...
0
votes
1
answer
297
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direct sum of two exact sequence is exact [duplicate]
recently I have had some problems thinking about this, does the direct sum of two exact sequence is also exact?
the picture is as follows:
all $M_i$ be R modules, and all map is R linear map,and the ...
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2
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152
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Coherence in closed monoidal categories
Let $(M, \otimes, I)$ be a left-closed (non-symmetric) monoidal category with left-internal hom $\underline{\operatorname{hom}}(-,-)$.
Denote by $\sigma_{A,B,C}: M(A\otimes B, C) \xrightarrow{\sim} M(...
- 3,108
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votes
0
answers
62
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Star-autonomous categories are linearly distributive categories with negation?
1.Context
On page 28 of Weakly distributive categories Cockett and Seely are trying to prove the following statement:
The notions of symmetric weakly distributive categories with negation and star-...
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vote
1
answer
180
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Prove that if $\alpha_{3}$ is an isomorphism, then the sequence is a short exact sequence.
I am having trouble making some connections in proving this result and I'd like some help sorting this out.
So here is the commutative diagram of $R$-modules and $R$-module homomorphisms.
Each row is ...
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1
answer
97
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If $E$ has pullbacks then $\text{SplitEpi}(E)$ has pullbacks
The book From groups to Categorical Algebra has as exercise to prove that if $E$ has pullbacks then the category of split epimorphisms of $E$ has pullbacks. This category consists of split ...
0
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1
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87
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$F$ left adjoint to $G \iff F,G$ define a functor from $\textbf{Arr}(\textbf{X}\times\textbf{A}) \to 2\times 1$ square CDs in $\textbf{Set}$?
Let $\textbf{A, X}$ be categories and $F:\textbf{X} \to \textbf{A}$ and $G: \textbf{A} \to \textbf{X}$. Then there is a map that takes an object in $\text{Arr}(\textbf{X}\times\textbf{A})$ (the arrow ...
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1
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1
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Showing an induced sequence is exact.
Edit: I got stuck in a couple of places in proving the statement below. Specifically:
$d$ is well-defined.
$\text{ker}(d) \subseteq \text{im}(\bar{v})$
$\text{ker}(\bar{v}’) = \text{im}(\bar{u}’)$
...
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1
vote
2
answers
158
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What is the $\textbf{Set}$-theoretical intuition behind understanding how power objects work in Topos Theory?
Definition. Let $\mathcal{E}$ be an elementary topos, $B \in \text{Ob}(\mathcal{E})$. Then a power object for $B$ is an object $PB \in \text{Ob}(\mathcal{E})$ together with a morphism $B \times PB \...
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