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Let $\mathcal A$ be an abelian category. Let $A$ be an object in $\mathcal A$ and let $(A_i)_{i\in I}$ be a set of subobjects of $A$. Then there is a subobject $\sum_{i \in I} A_i$ of $A$ which has all the $A_i$'s as a subobject, namely the image of the canonical morphism $\gamma \colon \coprod_{i \in I} A_i \to A$. Please refer to this question for more on this. However I am having trouble showing that this is the smallest such subobject.

What does "smallest subobject" mean in this sense? Does it mean given any subobject $X \longrightarrow A$ such that every $A_i$ is a subobject of $X$ then $$ \operatorname{im} \gamma \longrightarrow A \le X \longrightarrow A $$ OR does it mean that given any subobject $X \longrightarrow A$ such that every $A_i$ is a subobject of $X$ and also we have a commutative diagram of subobject arrows $$ \begin{array}{ccc} A_i & \rightarrow & A \\ & \searrow & \uparrow \\ & & X \end{array} $$ then $$ \operatorname{im} \gamma \longrightarrow A \le X \longrightarrow A ? $$

In the latter case I can see it because we can just use the universal property of the coproduct and the image to draw an arrow $\operatorname{im}\gamma \to X$ which works. However I cannot see this for the former case. If "smallest subobject" means the latter case why is it not completely specified in the language; is it a convention?

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up vote 2 down vote accepted

Let us say that $i:A\to X$ factors through $B$ if $j:B\to X$ is a subobject and there exists $k:A\to B$ such that $i=j\circ k$.

You are looking for a subobject $X\to A$ which has the following property: Whenever $A_i\to A$ factors through $Y$ for all $i$, then $X$ factors through $Y$ as well. In other words, whenever you have a subobject $Y$ "containing" all $A_i$, then $Y$ "contains" $X$. Now this property is satisfied if you take for $X$ the image of $\coprod_i A_i\to A$ simply by the universal property of the image.

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I misunderstood "contains $A_i$" to mean "have $A_i$ as a subobject." Thanks for clarifying. Is it necessarily true that $A_i$ will be a subobject of some $Y$ if it factors through $Y$ ? –  Paul Slevin Dec 17 '12 at 17:36
    
I have just realised that is true, because we are talking about subobjects and hence any factorisation must be a mono (and unique). –  Paul Slevin Dec 17 '12 at 18:10
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