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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Coherence in closed monoidal categories

Let $(C, \otimes, I)$ be a left-closed (non-symmetric) monoidal category with left-internal hom $\underline{hom}(-,-)$. Denote by $\sigma_{A,B,C}: C(X \otimes Y, Z) \xrightarrow{\sim} C(X,\underline{...
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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 \...
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Diagram chasing on borsuk ulam theorem proof

I'm trying to understand a proof on the Borsuk Ulam theorem, and it uses the fact that a continuous function from the sphere to the sphere induces a morphism on the Homology long exact sequences, as ...
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Compact support cohomology commute with direct limit

I'm not very familiar working with direct limits, since it's a new topic to me. The definition I'm currently using is that given a direct system $\left\lbrace A_\alpha\right\rbrace_{\alpha \in \Lambda}...
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If $n$ is even $\exists [f] \in \pi_{2n-1}(\mathbb{S}^n)$ such that $H(f) = \pm 2$

I'd like to prove that if $n$ is even $\exists [f] \in \pi_{2n-1}(\mathbb{S}^n)$ such that $H(f) = \pm 2$. Here $H(f)$ denotes the Hopf invariant of $f$. Sketch of the proof: Think $\mathbb{S}^n \vee \...
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How does a pushout and an exact sequence of modules give another exact sequence?

In this question and this one, it has been asked how an exact sequence of modules $0 \to K \to A \to F \to 0$ and the pushout of $P \leftarrow K \to A$ give rise to a second exact sequence, namely a ...
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Why does the inner square commute if all outer squares commute?

In proving a change of basis theorem in linear algebra, our professor draw this diagram and simply stated that because all the outer squares in this diagram commute, the inner square (green) must ...
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If three functions in commutative diagram are bijection, is the fourth one too?

Let's say we have a commutative diagram as in the following picture. The functions $f, h, g$ are all bijections. Can we conclude that $k$ is also a bijection? I need this as part of my proof, but in ...
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Short five lemma and a counterexample: commutative diagram

Let's say we have the following diagram $$\require{AMScd}\begin{CD} 0 @>>> A @>>> B @>>> C @>>> 0\\ {} @V{\alpha}VV @V{\beta}VV @V{\gamma}VV {} \\ 0 @>>> ...
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A colimit in a subcategory

I do not understand here what do the words colimit of a filtered diagram in $\cal P$,not just in $\cal F$ on the 2nd page mean, namely the word just there; is some direction trivial ? Namely,in ...
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Diagram Chase: Ascending Chain of Kernels

I have stumbled across what should be a routine diagram chase which I'm struggling with. The question is as follows: Let $f \colon A \to B$ be a map of Abelian groups. Let $p_A \colon A \to A$ and $...
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1 answer
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Looking for proof for variant of the Barratt-Whitehead lemma

$\require{AMScd}$Consider the $3×3$ commutative diagram with exact rows and columns: $$\begin{CD} @.@.0@.0@.0@.@.\\ @.@.@VVV@VVV@VVV@.@.\\ @.0@>>>X@>a>>X'@>a'>>X''@>>&...
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Exercise with exact sequence in an abelian category [duplicate]

Suppose (in an abelian category) to have two exact sequences, $0\to A\xrightarrow f B\xrightarrow g C\to 0$ and $0\to A'\xrightarrow {f'} B'\xrightarrow {g'} C'\to 0$, and three arrows $a:A\to A'$, $...
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Differentials in the spectral sequence associated to a double complex

Edit: I would very much like someone to comment on this. I have worked through Vakil's notes Spectral sequences: friend or foe? and I believe my description is correct. But there are so many indices ...
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Commutative Diagram of Composition and Addition (Chain Homotopies)

In my Homological Algebra class we have defined and studied a lot of things through commutative diagrams. However, when we got to defining chain homotopies, we had to add a descriptive statement ...
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Diagram chase proof regarding long exact sequence

$\newcommand{\im}{\operatorname{im}}$I am losing my sanity over this diagram chase. Given an abelian category and a cochain complex $A$, I am trying to prove that $\ker(A^i/\im d^{i+1} \to \ker d^{i-...
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About the proof of nine-lemma using snake lemma

I cannot follow the proof $(3\times 3)$ nine-lemma by using the snake lemma. Is there another way to understand the commutativity of the diagram after replacing the morphisms coming from the snake ...
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2 answers
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$\ker \varphi_p \subset (\ker \varphi)_p$ where $(\cdot)_p$ is taking the stalk of sheaves at the point $p$ (Diagram inside!).

I've already proven that $(\ker \varphi)_p \subset \ker \varphi_p$ using a commutative diagram and the definition $F_p = \lim\limits_{\longrightarrow \\ U \ni p} F(U) = \bigsqcup\limits_{U \ni p} F(U)/...
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Clarify about the connecting homomorphisms and the snake lemma

In class we said (without proof) that given a an exact sequence of chain complexes $$0\to \mathcal A\xrightarrow{f} \mathcal B\xrightarrow g \mathcal C\to 0$$ one can construct the long exact sequence ...
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2 votes
1 answer
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How to draw things like commutative diagram in PDE theory

The question maybe a little vague. I am working on PDE theory, I am trying to make my mind more clear by drawing some graph like commutative diagram. But the problem comes since in PDE theory u may ...
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Every representable presheaf is projective

Let $\mathcal{C}$ be a small category and let $\hat{\mathcal{C}}$ be its category of presheaves. I want to show that every representable presheaf $y_C\in \hat{\mathcal{C}}$ (for some $C\in\mathcal{C}$)...
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Showing that something is a Pushout

Hoping to get a better understanding of CW complexes in terms of pushouts, I have decided to take a look at Jeffrey Strom's Modern Classical Homotopy Theory (truly wonderful title for a book, by the ...
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2 votes
1 answer
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Showing commutativity of the ground monoid in a monoidal category

I have been trying to understand why the ground monoid in a monoidal category is commutative and every proof I have seen essentially uses the same thing to prove it namely by using the fact(apparently)...
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Can one obtain from the following diagram that the map $f_3$ is injective?

Let $A_i$ and $B_i$ be $R$-modules $(i=1,2,3)$. If in the diagram each map is $R$-linear, the rows are exact, both squares commute, and $f_1, f_2, \alpha_1, \beta_1$ are injective, is it possible to ...
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1 answer
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What is a clear, elegant proof that $\text{Hom}(\cdot, Y)$ is a right-exact functor in a category of modules?

If you search the site for this proof, you will find duplicates, however they are hard to understand. In other words they brush by the most critical points of the proof as if they were not worth ...
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1 answer
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Two homomorphisms $f\colon A\to A',\ g\colon B\to B'$ induce a homomorphism $f \otimes g: A\otimes B \to A'\otimes B'$

Yesterday I was working on an exercise-sheet, given the following definition For two Abelian groups $A$ and $B$ we define their tensor product $A\otimes B$ as the quotient of the free Abelian group ...
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1 answer
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Show that $\ker\sigma \subset \varphi_1(\ker\rho)$ and $\operatorname{im}\tau \subset \psi_2(\operatorname{im}\sigma)$.

Given the homomorphism of short exact sequences, I must show that $\ker\sigma \subset \varphi_{1}(\ker\rho)$ and $\operatorname{im}\tau \subset \psi_{2}(\operatorname{im}\sigma)$. For the first part, ...
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Proving simplicial homology is preserved on mapping?

Assume we have a the $\mathbb Z\text{ modules}$ $S1 \equiv (F, E, V)$ with boundary maps $(\partial_{FE}: F \rightarrow E$, $\partial_{EV}: E \rightarrow V)$, with the condition that $\partial_{EV} \...
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1 answer
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any cofibration $i:A \to B$ is a homeomorphism onto its image (question regarding the inverse map)

I was recently working on a problem that introduced the homotopy extension property as a cofibration $i:A \to B$. Let's say we are given the commutative diagram: Now, if $i:A \to B$ is the inclusion ...
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-1 votes
1 answer
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Suppose that $g$ is isomorphism. Then prove that $f$ is monic and $h$ is epic.

$$ \\ 0 \to A \to B \to C \to 0 \\ 0 \to A' \to B' \to C' \to 0 $$ These are exact and they occur a commutative diagram by homomorphism. $$ g=B\to B'\\ f=A\to A'\\ h=C \to C' $$ Suppose that $g$ is ...
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2 votes
1 answer
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$a \times \mathbf 1 \cong a$ in categories admitting products and having a terminal object $\mathbf 1$

I'm practicing my diagram chasing and reasoning skills, and, as an exercise, I'm trying to prove that if a category has products and also has a terminal object $\mathbf 1$, then for any $a$ an object ...
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1 vote
0 answers
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Category with multiple classes/types/labelled morphisms?

Is there a notion of a category with "labelled" or "typed" morphisms? I imagine that each morphism would belong to a certain class/type, and those classes would form a commutative monoid under ...
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3 votes
1 answer
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How should I proceed with diagram chasing in this commutative diagram?

From Rotman's Algebraic Topology: Consider a commutative diagram with exact rows: $\dots \rightarrow A_n \xrightarrow {i_n} B_n \xrightarrow {p_n} C_n \xrightarrow {d_n} A_{n-1} \rightarrow \...
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Doubt on a particular commutative diagram using the Tensor Product construction

I've posted two other questions* $[1]$ $[2]$, discussing and asking about the Tensor Product construction, in particular the "canonical construction", which is the one $[4]$: $$ \frac{F(V \times W)}{...
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2 votes
1 answer
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Creation of limits and diagram chasing

Let T be a monad in X. I want to prove that the forgetful functor G from category of T-algebras on X to X creates limits. I have read that it can be done via "diagram chasing". At this point, I am ...
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5 votes
1 answer
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Naturality of the ker-coker sequence of snake lemma in an abelian category

An important part of the snake lemma is the naturality of the ker-coker sequence produced by it. However, no source seems to state or prove this part for arbitrary abelian categories. However, it is ...
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2 votes
1 answer
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How to show that $a^1 \simeq a$ for all $C$-objects $a$ in a cartesian closed category? (only one direction of isomorphism proof is needed)

I have already proven that $a \times 1 \simeq 1 \times a \simeq a$ given a terminal object $1$ of a cartesian closed category $C$ (CCC). By CCC I mean that $C$ is finitely complete and has ...
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1 vote
1 answer
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In a topos, an internally functional relation induces a partial morphism

As a simple exercise, I wanted to write up a proof that in any topos $\mathcal{E}$, given a subobject $r:R\rightarrowtail X\times Y$ in $\mathcal{E}$, if $$(\forall x:X)(\forall y,z:Y)([R(x,y)\wedge R(...
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1 vote
1 answer
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Proving exactness not at the connecting hom in a Snake Lemma subproblem.

Here's a picture of the Snake Lemma from nLab: I'm having trouble showing exactness at $\ker g$ and haven't even got to the connecting hom yet. So let's focus on $\ker g$. Let $q : A' \to B'$ be ...
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1 vote
1 answer
58 views

A question about homotopy pushout

Let \begin{array}{ccc} X & \xrightarrow{} &Y \\ \downarrow & & \downarrow \\ Z & \xrightarrow{} & W\end{array} be a commutative diagram in a proper model catgeory and $P$ be ...
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1 vote
1 answer
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Injective map between cokers

Given a commutative diagram of vector spaces with exact rows I'd like to understand why knowing that $h$ is injective proves that the injection $ coker (\alpha) \to coker (\beta)$ is equivalent to $...
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1 vote
1 answer
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Making a diagram commute

Suppose we have a $k$-vector space $A$, two linear maps $$T_1, T_2:A\to A$$ and a bilinear map $\mu:A\otimes A\to A$. Is there a way to explicitly construct a map $F$ such that $$F\circ \mu = \mu\...
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1 vote
1 answer
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Limits via universal arrows and functor categories

I would like to understand is some detail the connection between the 2 snippets taken from McLane's book CWM. Namely, I do not follow the connection between the functor $S$ and categories and arrows $...
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  • 3,578
5 votes
0 answers
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Proving the four lemma using members

I am struggling with exercise VIII.4.2 in Mac Lane's Categories for the working mathematician. The exercise is about proving the four lemma about being an epimorphism using members in abelian ...
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pushout with zero upper map

If \begin{array}{ccC} A &\xrightarrow {0} & B\\ \downarrow & &\downarrow \\ C & \xrightarrow{g} & D \end{array} is a pushout. is $g$ zero map?
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