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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21 views

Question about the definition of chain homotopy

I recently learned about the definition of chain homotopy. If $f^\bullet, g^\bullet\colon C^\bullet\to D^\bullet$ are chain maps, then the definition is the following. A chain homotopy between $f^\...
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1answer
30 views

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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1answer
39 views

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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1answer
42 views

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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1answer
37 views

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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25 views

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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1answer
61 views

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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1answer
73 views

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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1answer
49 views

$a \times \mathbf 1 \cong a$ 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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39 views

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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1answer
38 views

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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46 views

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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1answer
40 views

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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1answer
71 views

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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1answer
51 views

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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1answer
35 views

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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1answer
31 views

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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1answer
39 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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1answer
24 views

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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1answer
31 views

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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1answer
40 views

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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0answers
117 views

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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0answers
11 views

Construction of diagram for given pushout

Let $\mathcal{C}$ be a proper model category and $P$ be the pushout of the diagram $Z \leftarrow X \rightarrow Y$ in $\mathcal{C}.$ Now consider $P' \in \mathcal{C}$ such that there is an weak ...
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12 views

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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4answers
79 views

Big list: collecting results of diagram chasing

I plan giving a homework problem on linear algebra for my students and my idea was to collect many results of diagram chasing. Could you help me and give me some results of diagram chasing you know? ...
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1answer
63 views

Composition of flat morphisms is flat

I do not understand in the snippet below why the composition of flat morphisms is flat. The full text (though in Czech) is given here. I do not understand these step given in bold, beginning from $q\...
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0answers
24 views

Existence of a chain map

Let $A$ be a module, $P$ be a chain complex of projective modules, $Q$ its subcomplex. Moreover, let $C$ be a resolution of the chain complex $A[0]$ (that is $(A[0])_0=A$ and $(A[0])_n=0$ otherwise; ...
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1answer
39 views

Prove categories equivalence using compositions of functors

Denote the diagram: $\require{AMScd}$ \begin{CD} C_{1} @>{G_{1}}>> C_2 @>{G_{2}}>> C_3\\ @VV{F_1}V @VV{F_2}V @VV{F_3}V\\ D_{1} @>{H_{1}}>> D_2 @>{H_{2}}>> D_3\\ \...
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0answers
22 views

When do equalizers preserve epis?

The question cannot be completely wrong, as it does not completely make sense, whence let me detail a bit. I'm in a monoidal category $(\mathcal{C},\otimes,\mathbb{I},\alpha,\lambda,\rho)$ with ...
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71 views

Strong monoidal equivalence from a symmetric monoidal category to a strict monoidal category

This question is related to point $(1)$ in this answer: https://math.stackexchange.com/a/190402/229776 Let $(\mathsf{C}, \otimes, 1, \alpha, l ,r, s)$ be a symmetric monoidal category, $(\mathsf{C}...
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1answer
166 views

Nine lemma prove using only diagram chasing

I want to prove the following lemma only by using diagram chasing: Nine Lemma: Let us have the following commutative diagram with exact columns and exact middle and bottom rows. So the top row is ...
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1answer
101 views

Good ol' Diagram Chasing fun - is my kernel finitely generated?

I'm in a bit of a pickle with a homework problem. Let $R$ be a commutative ring, and $M$ a finitely presented module (i.e. there exist $F_1$, $F_2$ free so that there is an exact sequence $A \to B \...
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1answer
48 views

what are the steps i have to take to find the sets that represent the red colored in the diagram?

i tried using the notation to make it all separate and then see if i can put all the solution i get from each a,b, c together but i couldn't understand what i am doing wrong. if anyone could help with ...
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1answer
120 views

$Q$ is an injective object in an abelian category iff $\text{Hom}_C(\cdot, Q)$ is exact, the $\Rightarrow$ direction.

In an abelian category, I understand that $\text{Hom}_C(X,Y)$ forms an abelian group for all $X, Y \in \text{Ob}(C)$. Thus to show that $Q$ is an injective object implies $\text{Hom}_C(\cdot, Q)$ is ...
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1answer
61 views

Promoting equivalence to adjoint equivalence: explicit construction of a unit and a counit

Let $F\colon\mathsf{C}\to\mathsf{D}, G\colon\mathsf{D}\to\mathsf{C}$ together with $\eta\colon 1_{\mathsf{C}}\to \mathsf{GF}, \epsilon\colon\mathsf{FG}\to 1_{\mathsf{D}}$ be an equivalence of ...
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92 views

Isn't “diagram” a misnomer in regards to the standard definition of being a functor $D : J \to C$?

Definition of diagram. When you draw an actual, what I call "diagram" on paper, it may have: Duplicated objects and arrows which is still valid. You don't always put in the compositions, as that ...
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68 views

Composition of a natural transformation with a functor

I have asked a similar question elsewhere but still in the snippet below, I don't know what is the functor $G$ and how differs $(G\ast\omega_F)$ from $(\omega_G\ast F)$ , what are they and what is $\...
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1answer
77 views

Consequence of five or four lemma?

I have two horizontal exact sequences of abelian groups. $\require{AMScd}$ \begin{CD} 0 @>{}>> A_1 @>{}>> A_2 @>{}>> A_3 @>{}>> A_4 @>{}>> 0\\ @| @VeVV @...
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0answers
85 views

Cute/striking application(s) of Snake Lemma outside homological algebra

When you teach algebra to students, it's often easy to find cute/direct applications of "big" theorems to motivate how useful these results can be. For example, group actions, Sylow theorems, or the ...
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1answer
37 views

Commutativity of Diagram in Top “Lift-like property but for covers”

Let $\psi,\chi,\phi$ be continuous functions, with $\psi,\xi$ surjective. Does there necessarily exists a continuous function $\eta$ such that the diagram commutes in Top: $$\require{AMScd} \begin{CD}...
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1answer
52 views

WIth pushout each orthogonality class is an injectivity class

I cannot see how follows the equality $\cal M^{\bot}=M^*$-Inj below: (both inclusions are wanted) where $X \in {\cal H}^\bot \iff \forall f\in{\cal H}: \mathrm{Hom}(f,X)$ bijective and $X \in {\...
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2answers
71 views

What does the notations $X \times_{Z} Y$ mean?

Right now I am learning how to create commutative diagrams in Latex, and since I am also studying category theory right now, I have come across the notation before in category theory texts like ...
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1answer
46 views

Equivalent definition of Cofibration

Basic question from may's concise course about cofibration. In the beginning of chapter 6 (search pg 51 in this pdf: https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf), May gives two ...
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1answer
61 views

Show that this mapping between localized modules is an isomorphism

Let $R$ be a ring. Let $M$ and $N$ be $R$-modules where $M$ is finitely presented. Then for every multliplicative set $S \subset R$ the canonical mapping $~~~~~~~~~~~~~~~~~~~~~~~~~~~~Hom_R(M,N) \...
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1answer
17 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
36 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 \...
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1answer
94 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:\...
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1answer
245 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 $\mathcal{...
2
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1answer
62 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 $a_{1}$, ...
4
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1answer
247 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 ...