# On morphisms in an abelian category

$$\underline {Background}$$: Suppose ,we are in an abelian category $$\mathcal C$$ and let $$B \in \mathcal C$$ be an object.

Let,$$f_1,f_2 \in Hom(B,B)$$.

Since $$\mathcal C$$ is an abelian category $$f_1-f_2 \in Hom(B,B)$$.

$$\underline {Question(1)}$$:what is the image of $$f_1-f_2$$?

i.e can we calculate this image in general in terms of kernel ,cokernel ,image of the morphisms $$f_1$$,$$f_2$$?

$$\underline { Guess}$$: may be under some suitable condition it could be $$Im(f_1)/Im(f_2)$$ (the only natural thing that comes to my mind is this even though I do not see why Im$$f_2$$ should be embedded in Im$$f_1$$).But I do not see any clue how to proceed further.

$$\underline {Question (2)}$$: even if we cannot represent it generally in terms of kernel,cokernel and image of $$f_1,f_2$$ how to think of this image category theoretically,i.e diagrammatically.

Any help from anyone is welcome.

• Why exactly do you think there should be something like this in Question 1? I mean, if I encounter such a question, the first thing I would do is to see if this holds for abelian groups or vector spaces over some field. – Paul K Dec 16 '18 at 16:47

## 1 Answer

Question 1: You cannot say anything about the image of $$f_1-f_2$$. Here is an example. Let $$\mathcal{C}$$ be the category of real vector spaces and $$B = \mathbb{R}^n$$. The set $$\text{End}(\mathbb{R}^n)$$ of endomorphisms $$\phi : \mathbb{R}^n \to \mathbb{R}^n$$ is a normed linear space and it is well-known that the set $$\text{GL}(\mathbb{R}^n)$$ of automorphisms is open in $$\text{End}(\mathbb{R}^n)$$.

Let $$f_1 = id_B$$ and $$V \subset \mathbb{R}^n$$ any linear subspace. Choose any linear map $$g : \mathbb{R}^n \to \mathbb{R}^n$$ such that $$\text{im}(g) = V$$. Hence for sufficiently small $$\epsilon > 0$$ the linear map $$f_2 = id_B + \epsilon g$$ belongs to $$\text{GL}(\mathbb{R}^n)$$.

Both $$f_1, f_2$$ are isomorphisms, thus their kernels and cokernels are trivial and their images are $$\mathbb{R}^n$$. However, $$f_1 - f_2 = - \epsilon g$$ whose image is $$V$$.

Question 2: I do not think that one can get a reasonable answer.