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Let C abelian category.

And let $\{M_{\lambda},f_{\nu,\lambda}:M_\lambda\rightarrow M_\nu\}$, direceted system of C indexed by I and $\lim{M_\lambda}$ be direct limit of $\{M_{\lambda} ,f_{\nu,\lambda}\}$

Let $\phi_\lambda$ be canonical homomorphism from ${M_\lambda}$ to $\lim{M_i}$.

Then, is $Ker\phi_\lambda$ isomorphic to $\sum_{\lambda\leq \nu}\ Kerf_{\nu,\lambda}$

This is categorical interpretation of the proposition below.

$\forall x\in M_\lambda,(\phi_\lambda(x)=0\Leftrightarrow\exists\nu\geq\lambda,f_{\nu\lambda}(x)=0)$

This is true in Categories of Abelian group. (See https://stacks.math.columbia.edu/tag/00D7)

If it's not true, what's the "minimum condition" where it holds? I think it may be related to Grotandiek AB condition.

I aim to prove the proposition above categorically." I also watched this question. Abelian categories and axiom (AB5)

George
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The statement you presented is not generally true. The kernel of the natural homomorphism πλ from Mλ to the direct limit limMi is not isomorphic to the sum (∑λ≤ν Kerfν,λ).

To see why, let's consider a counterexample. Take the following simple case:

Consider the abelian category C to be the category of abelian groups, and let I be the set of natural numbers N. For each n in N, let M_n be the abelian group Z/nZ, and for each pair (m, n) in N × N such that m divides n, let f_m,n be the canonical inclusion homomorphism from Z/mZ to Z/nZ.

The direct limit of this directed system is the abelian group limMi = Z, where the direct limit map fν: Mν -> limMi is the canonical projection map for each n in N.

Now, let's calculate the kernel of π_n, where n is a natural number. Since the direct limit is Z, the kernel of π_n is the subgroup of Z that maps to zero in Z/nZ. It is easy to see that the kernel of π_n is isomorphic to the direct sum (∑m|n Z/mZ).

On the other hand, consider the sum (∑m|n Kerf_m,n). Each Kerf_m,n is isomorphic to Z/mZ, so (∑m|n Kerf_m,n) is isomorphic to the direct sum (∑m|n Z/mZ).

In general, the two subgroups are not isomorphic. For example, if we take n = 6, then the kernel of π_6 is Z/6Z, but the sum (∑m|6 Z/mZ) is Z/2Z ⊕ Z/3Z, which is not isomorphic to Z/6Z.

Therefore, the statement you provided is not true in general.

Regarding your question about the "minimum condition" where the statement holds, I'm not aware of a specific condition that guarantees the isomorphism you mentioned. It may be worth investigating specific properties of the directed system, such as the Grothendieck AB condition or other coherence conditions, to see if they lead to the desired result. However, it is important to note that such conditions might impose additional restrictions on the directed system or the category C.

I encourage you to further explore the topic and consult relevant literature on directed limits and coherence conditions in abelian categories for a more comprehensive understanding.

Tabeye
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