# 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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### 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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### 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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### $a \times \mathbf 1 \cong a$ 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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### 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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### 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&...