Questions tagged [diagram-chasing]

For questions about proofs using equivalent map compositions in commutative diagrams in homological algebra, or in category theory in general.

13 questions
258 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 ...
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 ...
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 $R$-...
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 &...
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 ...
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 ...
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-...
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 ...
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 ...
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 ...
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 ...
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?