136
votes
Short exact sequences every mathematician should know
There is one obvious sequence that underlies all vector analysis and a lot that builds up on it, no matter if its applied analysis, PDE, physics or the original foundations of algebraic topology. Yet ...
89
votes
Accepted
When size matters in category theory for the working mathematician
Very often one has the feeling that set-theoretic issues are somewhat cheatable, and people feel like they have eluded foundations when they manage to cheat them. Even worse, some claim that ...
- 8,186
76
votes
Short exact sequences every mathematician should know
This might be very basic, but the short exact sequence
$$
0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0
$$
is both an injective resolution of $\mathbb{Z}$, and a flat resolution of $\...
75
votes
Short exact sequences every mathematician should know
The exponential sheaf sequence:
$$0\to 2\pi i\,\mathbb Z \to \mathcal O_M {\buildrel\exp\over\to}\mathcal O_M^*\to 0$$
where $\mathcal O_M$ is the sheaf of holomorphic functions on the complex ...
67
votes
Short exact sequences every mathematician should know
I find it hard to believe that three days have gone by and no one has explicitly mentioned
$$
0 \to \Bbb Z \to \Bbb R \to \Bbb S^1 \to 0
$$
62
votes
Accepted
What is homology anyway?
Let's take coefficients in a field $k$, for simplicity.
On 2): the singular cohomology of a topological space $X$ is the dual of its singular homology, almost by definition. But if $X$ is a space for ...
- 17.6k
61
votes
Short exact sequences every mathematician should know
I think a short exact sequence that every teacher should know is
$$ 0 \to \mathbb R^d \to \mathrm{Isom}(\mathbb R^d) \to \mathrm{O}(\mathbb R^d) \to 0, $$
maybe for $d=2$ or $d=3$. Better still, ...
53
votes
whence commutative diagrams?
I can muddy the waters...!
According to editor E. Scholz of Hausdorff’s Collected Works (2008, p. 884):
In a note of 3/20/1933 (Nachlass, fasc. 449) and in a further undated note (fasc. 571), ...
- 29.5k
48
votes
Serre's theorem about regularity and homological dimension
In 1953-54 E. Artin asked me to describe homological algebra to him. In addition to Ext and Tor, I decided to show him the "homological proof" of the Hilbert Basis Theorem and indicated that the same ...
- 481
47
votes
Short exact sequences every mathematician should know
The short exact sequence
$$ 0 \to \mathrm{rad}({\mathfrak g}) \to {\mathfrak g} \to {\mathfrak g}/\mathrm{rad}({\mathfrak g}) \to 0$$
separates a Lie algebra ${\mathfrak g}$ into its solvable radical $...
47
votes
Accepted
Why stable $\infty$-categories?
I already answered some version of this question in this answer, but let me try to expand a bit on the concrete advantages in mathematical practice. For understanding the following you need to take on ...
- 16.2k
46
votes
Short exact sequences every mathematician should know
"Every mathematician should know" is too much to ask, but I do think the following is a great short exact sequence that captures a vital phenomenon:
$$0 \to K(H) \to B(H) \to Q(H) \to 0.$$
$...
45
votes
Short exact sequences every mathematician should know
I strongly doubt there is any short exact sequence that every mathematician should know, but I certainly wish that those of them who know that for a (co)chain complex $(C,d)$
$$
0\to\operatorname{Im}(...
41
votes
Short exact sequences every mathematician should know
An example of a short exact sequence satisfying your first desiderata, but one which you probably won't fully understand till you are further along in homological algebra, is the Universal Coefficient ...
36
votes
Accepted
Replacing triangulated categories with something better
My opinion, and that of many other people although not of everyone, is that the "correct" notion is that of stable ∞-category.
Now, this is not a category in the strictest sense, rather a ...
- 16.2k
35
votes
Short exact sequences every mathematician should know
Within the category of Banach spaces and bounded linear maps,
$$0\to c_0 \to \ell_\infty \to \ell_\infty / c_0 \to 0$$
is a paradigm example of a short exact sequence that does not split, contrary to ...
34
votes
whence commutative diagrams?
Eduard Study in Von den Bewegungen und Umlegungen, Math. Ann. 39 (1891) 441-566, writes on p. 508:
Here $g, g^*, g'$ are rays in space with polar planes $\gamma, \gamma^*, \gamma'$, $\mathfrak P$ is ...
- 29.5k
29
votes
intuition for hochschild homology
Slogan: Hochschild homology is a (derived) categorification of the trace.
This means the identity at the end of John Pardon's answer is a categorification of the identity $\text{tr}(AB) = \text{tr}(...
- 115k
29
votes
What should I call a "differential" which cubes, rather than squares, to zero?
I would just call it a module over the truncated polynomial algebra $k[D]/D^3$. Your two flavors of homology appear as positive odd-degree and even-degree groups in $\operatorname{Ext}^*_{k[D]/D^3}(k,...
29
votes
Short exact sequences every mathematician should know
How about the short exact sequence that expresses that every group can be expressed in terms of generators and relators? For any group $G$, there is a short exact sequence (in fact many) of the form $$...
29
votes
When size matters in category theory for the working mathematician
Here's an example that links more to mathematical practice outside category theory proper. Recall that for a small site $(C,J)$, where I take $J$ to be a Grothendieck pretopology on the small category ...
- 33.9k
28
votes
Short exact sequences every mathematician should know
For any abelian group $A$, there is a short-exact sequence $$0 \to T(A) \to A \to A/T(A) \to 0,$$
where $T(A)$ is the torsion subgroup of $A$, and $A/T(A)$ is torsion-free.
27
votes
What is homology anyway?
For a long time (and still today), I very much shared the confusion of the OP. I think Jacob Lurie gives a very clear take on the standard perspective, but Mike Shulman does have a very valid ...
- 19.9k
27
votes
Short exact sequences every mathematician should know
The kernel-cokernel exact sequence: in an abelian category, given $A \xrightarrow{f} B \xrightarrow{g} C$, the following sequence is exact
$$ 0 \to \ker f \to \ker gf \to \ker g \to \text{coker } f \...
27
votes
Accepted
Breen-Deligne packages and the liquid tensor experiment
The comments have already given the answers, but let me assemble them here with my account of the story.
When Scholze first posted the Liquid Tensor Experiment, it was quickly identified (by both ...
- 5,444
26
votes
Accepted
About the abelian category of endofunctors of $\mathsf{Vect}$
There are a couple of equivalent ways to characterise polynomial functors. One is to say that $F$ is a polynomial functor of degree $n$ if the function $$\hom(U, V)\to \hom(F(U), F(V))$$ is polynomial ...
- 10.7k
26
votes
What is homology anyway?
I generally think about the relationship differently than Jacob, probably because I'm coming from an algebraic topology background rather than an algebraic geometry one. I would say that if $\mathcal{...
- 65.1k
26
votes
Accepted
Existence and uniqueness of Haar measure on compacta; a cohomological approach
Fix a compact group $G$ and consider its category of Banach representations:
the objects are (complex) Banach spaces $X$ endowed with a $G$-action by automorphims (not necessarily isometries) such ...
- 11.4k
26
votes
Short exact sequences every mathematician should know
Given a finitely generated module $M$ over a commutative Noetherian ring $R$, there is a short exact sequence $$0\to M_1 \to R^n \to M\to 0$$
where you map $1$ in each $R$ to a generator of $M$ and $...
26
votes
When size matters in category theory for the working mathematician
The other answers are good, but I would like to point out that Ivan's "uncheatable" lemma can in fact be cheated. The proof of that lemma (due to Freyd) makes inescapable use of classical ...
- 65.1k
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