exact sequences in concrete abelian categories

Stream: condensed mathematics

[Johan Commelin]

Suppose I have a concrete abelian category $𝒜$ and a short complex $A \xrightarrow{f} B \xrightarrow{g} C$ in $𝒜$. Under what conditions can I conclude that the complex is exact by checking $∀ b, g(b) = 0 → ∃ a, f(a) = b$? I guess the pseudo-elements API allows to do this quite generally. But here I'm explicitly interested in making the connection with actual elements.

[Johan Commelin]

Does this work in every concrete abelian category?

[Adam Topaz]

How is concrete defined here? Do you want a forgetful functor to Ab?

[Johan Commelin]

Hmm, good point. I guess so?

[Adam Topaz]

Certainly preserving kernels and cokernels would suffice.

[Adam Topaz]

and you want the functor to be additive presumably.

[Johan Commelin]

Adam Topaz said:

Certainly preserving kernels and cokernels would suffice.

Right, my question is whether there are pathological examples where this breaks down

[Kevin Buzzard]

So there's three conflicting definitions of "concrete abelian category" here -- is that the conclusion? One is "concrete and abelian", one is "faithful functor to Ab", one is "faithful functor to Ab which preserves kernels and cokernels"

[Reid Barton]

What you really want is a conservative functor that preserves kernels and cokernels (since you want to know whether the induced map $\mathrm{coker}\, f \to \ker g$ is an isomorphism)--but I guess this is one of those funny coincidences in abelian categories:

• if $F$ is faithful then it reflects "being the zero object", since $X = 0$ iff $\mathrm{id}_X = 0$;
• then if $F$ also preserves kernels and cokernels then it reflects isomorphisms, because $f$ is an isomorphism iff its kernel and cokernel are the zero object