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It seems to me that there are (at least) two notions of exact sequences in a category:

1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite of two adjacent maps is zero) $$\dotsc \to A_0 \to A_1 \to A_2 \to \dotsc$$ exact if the canonical map $\mathrm{im}(A_i \to A_{i+1}) \to \ker(A_{i+1} \to A_{i+2})$ is an isomorphism for all $i$.

2) Let $\mathcal{C}$ be a pointed category with kernels and cokernels. Then we call a complex $$0 \to A \to B \to C \to 0$$ exact if $A \to B$ is a kernel of $B \to C$ and $B \to C$ is a cokernel of $A \to B$.

(There is also the notion of an exact category, but my question is not about such extra structures on a category.)

Surely 1) is well-known, and 2) appears as Definition 4.1.5 in Mal'cev, protomodular, homological and semi-abelian categories by Borceux and Bourn (are there other references?). I think that 2) generalizes to long exact sequences in a natural way, because we can slice them into short exact sequences and take this as a definition; is there a reference for this? But my actual question is about the equivalence between 1) and 2).

It is well-known that 1) and 2) are equivalent for abelian categories, but 2) is stronger than 1) in the category of pointed sets. Namely, $$0 \to (X,x_0) \xrightarrow{f} (Y,y_0) \xrightarrow{g} (Z,z_0) \to 0$$ is exact in the sense of 2) if and only if $f$ is injective, $g$ is surjective, and $\mathrm{im}(f)=\ker(g)$, whereas for 1) we only need that $f$ is weakly injective, meaning that $\ker(f)=0$.

I have to admit that this appears to be a bit strange for me. In particular, it means that 1) is not a self-dual notion, since $(Y,y_0) \xrightarrow{g} (Z,z_0) \to 0$ is exact iff $g$ is surjective iff $g$ is an epimorphism, but $0 \to (X,x_0) \xrightarrow{f} (Y,y_0)$ is exact with 1) iff $f$ is weakly injective, which does not imply that $f$ is a monomorphism, i.e. injective.

So why is 1) the standard definition of an exact sequence of pointed sets? Actually I only know one prominent example, namely the long exact sequence of homotopy groups (sets in degree $0$) associated to a fibration, but this doesn't start with $0$; it ends with $0$, so that there is no problem. Are there examples which show that 1) is more useful than 2) for pointed sets? (If this is true at all.)

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    $\begingroup$ Don't you answer your own question? The definition reflects what can be proved in the only useful example. $\endgroup$ – Alex Degtyarev Apr 28 '15 at 21:15
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    $\begingroup$ The canonical map in 1) doesn't exist unless you assume that the composition of two consecutive arrows is zero, you also need the category to be pointed in order to define kernels. 2) doesn't generalize to long exact sequences unless images exist. The long exact sequence of homotopy groups has much more structure in low dimensions. There is a group acting on a set, the images of the following map of sets correspond to orbits, etc. $\endgroup$ – Fernando Muro Apr 28 '15 at 22:00
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    $\begingroup$ Incidentally, Martin, if I can go off-topic: I had the impression that the Bourn--Borceux framework is tailored to categories which are like abelian categories but not additive; while Prosmans / Schneiders is tailored to additive categories that aren't abelian but somehow aren't too far from abelian. Do you know any source which simultaneously extends both concepts? (the category I have in mind is that of Banach spaces and "short" linear maps) $\endgroup$ – Yemon Choi Apr 28 '15 at 22:22
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    $\begingroup$ The opposite of the category of pointed sets is semi-abelian, so the Borceux–Bourn definition is very much applicable. $\endgroup$ – Zhen Lin Apr 28 '15 at 23:26
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    $\begingroup$ So far as the final question, (1) is the standard definition because this is what appears for the long exact sequence, in particular because the basic structure of the exact sequence changes with each basepoint when the fundamental group changes. We can't demand something as strong as (2) because that's not the structure that we see. $\endgroup$ – Tyler Lawson Apr 28 '15 at 23:47
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The basic reason for the appearance of 1) in the long exact sequence in homotopy is that it is exactly the kind of exactness you get if you apply $\pi_0$ to a fiber sequence $F \to E \to B$ of pointed spaces. You simply don't get 2) because $F$ cannot see what happens outside of the connected component of the basepoint of $B$.

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I am not sure if this answers the question explicitly, but in terms of references you should look up the notion of fibration of groupoids which I think first appeared in an exercisein the 1968 edition of what is now Topology and Groupoids, and is well discussed and used in that book.

The family of exact sequences derived from a fibration of groupoids is an algebraic model of the bottom end of the exact sequences of a fibration of spaces. These sequences have been used by P.R. Heath in discussing Nielsen fixed point theory: Topology and its Applications 181 (2015) 3–33. See also arXiv:1207.6404.

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    $\begingroup$ I fail to see what this has to do with the question. $\endgroup$ – Andy Putman May 30 '15 at 18:10

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