# Universality of Ext functor using Yoneda extensions

Theses are simple and natural questions, but I could not find anything about it. If anyone has an answer or a reference this would be very much appreciated.

Let $\mathcal{C}$ be an abelian category (possibly without enough injective nor projective).

(i) Let $A,B \in \mathcal{C}$. When are the $\mathrm{Ext}^n(A,B)$ (defined using Yoneda extensions) sets ?

(ii) Let $A \in \mathcal{C}$ and suppose that $\mathrm{Ext}^n(A,B)$ is a set for all $B \in \mathcal{C}$. Is $\mathrm{Ext}^\bullet(A,-)$ a $\delta$-functor ? If yes is it universal ?

(iii) Same as (ii) in the special case where $\mathcal{C}$ has enough projective.

PS : I edited a bit the question in view of Fernando Muro's comments.

• It is surely a $\delta$-functor, as it comes with a long exact sequence associated to each short exact sequence, with the connecting morphism given by Yoneda product by the short exact sequence itself. – Mariano Suárez-Álvarez Feb 8 '13 at 3:13
• What do you mean by 'universal'? – Fernando Muro Feb 8 '13 at 7:45
• @Fernando: en.wikipedia.org/wiki/Delta-functor – Martin Brandenburg Feb 8 '13 at 10:25
• @Mariano : Proving that the long sequence is still exact without the derived functor property does not seem obvious to me (I see that one can still define the long sequence, using Yoneda product). – Arkandias Feb 8 '13 at 13:24
• @Arkandias, that would be another question that I'd be very happy to answer. Concerning Ext being sets, Verdier didn't possibly show it since in general it is not true. This is also related to the fact that not every abelian category has a derived category. – Fernando Muro Feb 10 '13 at 23:47

In their paper entitled "Extension categories and their homotopy", Neeman and Retakh define a spectrum of extensions $\operatorname{Ext}(A,B)$ for any two objects in an exact category $\mathcal E$ such that $\pi_{-n}\operatorname{Ext}(A,B)=\operatorname{Ext}_{\mathcal E}^n(A,B)$, in the sense of Yoneda, for any $n\geq 0$. Positive-dimensional homotopy groups vanish. The spectrum $\operatorname{Ext}(A,B)$ is an $\Omega$-spectrum defined by the classifying spaces of the categories $\operatorname{Ext}^n(A,B)$ of $n$-fold Yoneda extensions.

Given a short exact sequence $B\hookrightarrow C\twoheadrightarrow D$, Quillen's Theorem B shows that the homotopy fiber of $\operatorname{Ext}^n(A,C)\rightarrow \operatorname{Ext}^n(A,D)$ is $\operatorname{Ext}^n(A,B)$, $n\geq 0$. Hence, for spectra, the homotopy fiber of $\operatorname{Ext}(A,C)\rightarrow \operatorname{Ext}(A,D)$ is $\operatorname{Ext}(A,B)$. The long exact sequence on homotopy groups defines now a $\delta$-functor $\operatorname{Ext}^\bullet(A,-)$.

Universality follows from Yoneda's lemma. If $T$ is another $\delta$-functor, a natural transformation $\operatorname{Hom}(A,-)=\operatorname{Ext}^0(A,-)\rightarrow T^0$ extends uniquely to a morphism of $\delta$-functors $\operatorname{Ext}^n(A,-)\rightarrow T^n$, $n\geq 0$, as follows. An $n$-fold extension $B\hookrightarrow X_1\rightarrow\cdots\rightarrow X_n\twoheadrightarrow A$ factors as the 'composition' of short exact sequences $$Y_{n-1}\hookrightarrow X_n\twoheadrightarrow Y_n$$ with $Y_0=B$ and $Y_n=A$. In particular we obtain morphisms $$T^0(A)\rightarrow T^1(Y_{n-1})\rightarrow T^2(Y_{n-2})\rightarrow\cdots\rightarrow T^{n-1}(Y_1)\rightarrow T^n(B).$$ The image of the previous extension by extension by $\operatorname{Ext}^n(A,B)\rightarrow T^n(B)$ is the image by this composite of the element in $T^0(A)$ classifying the natural transformation we started with (via Yoneda's lemma). Everything is well defined by the properties defining a $\delta$-functor.

This is only a sketch of proof. If you intend to use it in a paper you should probably provide some details at some points, e.g. carefully check the hypotheses of Quillen's Theorem B.

• While the paper by Neeman and Retakh is very nice, this can be done by hand. The existence of the long exact sequence (with connecting maps given by Yoneda product with the short exact sequence) is, for example, an exercise in the book by Hilton and Stammbach. – Mariano Suárez-Álvarez Feb 16 '13 at 6:27
• Neeman-Retakh's paper is also done by hand. Besides, using spectra gives you the long exact sequence for free, you get a tedious checking spared. – Fernando Muro Feb 16 '13 at 8:41
• See B.Mitchell "Theory of Categories\$ cpa. VII + Cor. 2.3 p.198. – Buschi Sergio Feb 16 '13 at 12:10
• Vladimir Retakh informs me that his paper with Neeman mentioned above grew up from his paper Retakh, V. S. Homotopy properties of categories of extensions. (Russian) Uspekhi Mat. Nauk 41 (1986), no. 6(252), 179–180. (English translation: Russian Math. Surveys 41 (1986), no. 6, 217–218.) He was trying to define Massey products without using resolutions. This can be regarded as a step forward in the direction of this question. – Fernando Muro Jul 19 '18 at 23:19
• Another approach to Massey products by the same author can be found in Retakh, Vladimir S. Opérations de Massey, la construction S et extensions de Yoneda. C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 11, 475–478. Here, the S-construction is Waldhausen's approach to K-theory. At a first glance, it looks like a very short and well written paper. I'll definitely dive into it as soon as I can. – Fernando Muro Jul 19 '18 at 23:19