# Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

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### Subcategory determined by composition series

Suppose $A$ is an artin algebra and take the category $\operatorname{mod}A$ of finitely generated $A$-modules. Consider the following construction. Let $M$ be an $A$-module. Since $A$ is artinian, $M$ ...
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### What are some examples of injective sheaves?

Injective objects in the category of abelian groups are precisely the divisible groups. However, despite using injective resolutions of sheaves everywhere, I realized that I did not know a single ...
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### Proving that there is a functor $F: Grp \rightarrow Ab$ s.t $F(G)=G_{ab}$- How to deal with the quotient?

Let $G$ be a group. $f$ a group hom. $f:G\rightarrow H$ I define: $F(G)=G_{ab}$ and $F$ : $G$ $\rightarrow$ $G_{ab}$ a homomorphism, where $G_{ab} : = G/[G,G]$ is the abelianization of group $G$....
I am new to Category Theory and it was recommended to me to start my understanding of basic concepts by reading $\textit{Basic Category Theory}$ by Tom Leinster. But I am finding myself struggling a ...
### How do I complete the proof of the composition rule for the functor that sends a ring $R$ to its group of units $R^×$?
I have to prove that there is a functor Ring $\rightarrow$ Grp that sends a ring $R$ to its group of units $R^×$. In order to to that : Let $R,S,T$ be rings, $h$ a homo of rings from $R$ to $S$ and I ...