# 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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### 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 $t$ ...
1 vote
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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 \...
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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 \...
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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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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 \... • 1,080 0 votes 0 answers 75 views ### 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 2 votes 1 answer 84 views ### 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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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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### 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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### 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 ...
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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 ...
1 vote
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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 ...
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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 ...
1 vote
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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'@&...
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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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### 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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### 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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### 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 ...
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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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### 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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### 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 \...