homological algebra - The sharp 3x3 lemma: a proof by universal properties?

I was reading this paper a while ago, and I couldn't figure out how to prove a lemma that was left as an exercise by only using universal properties and the definition of an abelian category.

I'll reproduce the diagram:

$$ matrix{
&&0&&0&&0
cr&&downarrow&&downarrow&&downarrow
cr &&A_1 & & B_1& &C_1
cr &&downarrow & &downarrow&&downarrow
cr &&A_2 & to & B_2 & to & C_2 &
cr &&downarrow &&downarrow&&downarrow
cr 0&to&A_3 & to & B_3 & to & C_3
} $$

With all rows and columns exact. (This diagram lives in an abelian category).

Show that there exists an exact sequence $A_1to B_1to C_1$ making the diagram commute.

Sure, it's not too hard with elements, I mean, it's just part of the snake lemma. However, proving it with universal properties is another story. By the universal property of the kernel, there are natural maps $A_1 to B_1$, and $B_1 to C_1$. Proving that this is exact is another story entirely. I believe I was able to show (I tried this a few months ago) that the top left corner (not counting zeros) is cartesian (a pullback square), but I still couldn't prove exactness.

I repeat, this is for a proof without elements. It should rely only on the definition of an abelian category and universal properties. If you happened not to click the link to the paper, the whole point is a proof without elements. I'd really like to see at least one proof in homological algebra actually done from the definition, just because it would be extremely instructive.

qa.quantum algebra - Are the “identity object axioms” in the definition of a braided monoidal category needed? (Answered: No)

I am asking because the literature seems to contain some inconsistencies as to the definition of a braided monoidal category, and I'd like to get it straight. According Chari and Pressley's book ``A guide to quantum groups," a braided monoidal category is a monoidal category $mathcal{C}$ along with a natural system of isomorphisms $sigma_{U,V}: U otimes V rightarrow V otimes U$ for all pairs of objects $U$ and $V$, such that

(i) The ``Hexagon" axioms (two commutative diagrams) hold.

(ii) The ``identity object" axioms: $rho_V= lambda_V circ sigma_{{bf 1},V}: {bf 1} otimes V rightarrow V$
and
$lambda_V= rho_V circ sigma_{V, {bf 1}}: {V} otimes {bf 1} rightarrow V$,
where $lambda_V$ and $rho_V$ are the isomorphisms of $V otimes {bf 1}$ and ${bf 1} otimes V$ with $V$ that are part of the definition of monoidal category. See Chari-Pressley Definitions 5.2.1 and 5.2.4. They use the term "quasitensor category," but note on p153 that the term "braided monoidal category" is equivalent.

However, in some references (ii) seems to have been dropped. I am thinking in particular of Definition 3.1 is this
expository paper,
and the
wikipedia article.
The wikipedia article goes further, and suggests that (ii) somehow follows from (i) and the axioms of a monoidal category. So, my questions are.

1) Is (ii) needed? That is if we do not impose (ii), does it follow from (i) and the axioms of a monoidal category?

2) If (ii) is needed, can someone provide an example demonstrating why? That is, provide an example of a monoidal category $mathcal{C}$ along with maps $sigma_{U,V}$ such that (i) holds but (ii) fails. Alternatively, if (ii) is not needed, I'd like a proof (or reference to a proof) that it follows from other axioms.

cv.complex variables - Restriction of a complex polynomial to the unit circle

I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).

Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there is the following dichotomy. Either we can write $f(z)=g(z^k)$ for some other polynomial $g$ and some integer $k>1$, or the restriction of $f(z)$ to the unit circle is a loop with only finitely many self-intersections. (Which means, more concretely, that there are only finitely many pairs $(z,w)$ such that $|z|=1=|w|$, $zneq w$ and $f(z)=f(w)$.)

EDIT. Here are a couple reasons why I believe the statement is correct.

1) The statement is equivalent to the following assertion. Consider the set of all ratios $z/w$, where $|z|=1=|w|$ and $f(z)=f(w)$ (here we allow $z=w$). If $f$ is a nonconstant polynomial, then this set is finite.

[[ Here is a proof that the latter assertion implies the original statement. Suppose that there are infinitely many pairs $(z,w)$ such that $|z|=1=|w|$, $zneq w$ and $f(z)=f(w)$. Then some number $cneq 1$ must occur infinitely often as the corresponding ratio $z/w$. However, this would imply that $f(cz)=f(z)$ (as polynomials). It is easy to check that this forces $c$ to be a root of unity, and if $k$ is the order of $c$, then $f(z)=g(z^k)$ for some polynomial $g(z)$. ]]

Going back to the latter assertion, note that the set of all such ratios is a compact subset of the unit circle, and it is not hard to see that 1 must be an isolated point of this set. So it is plausible that the whole set is discrete (which would mean that it is finite).

2) If I am not mistaken, experiments with polynomials that involve a small number of nonzero monomials (such as 2 or 3) also confirm the original conjecture.

at.algebraic topology - H-space structure on infinite projective spaces

There's also a different way of writing down the $H$-space structure, that I like for its algebro-geometric flavor. (I'll talk about $mathbb{C}P^infty$ here, and $mathbb{R}P^infty$ should be analogous.)

Regarding $mathbb{C}P^infty$ as a classifying space for complex line bundles, we know that this $H$-space structure is supposed to implement "tensor product of line bundles". In a (not very explicit) sense this tells us the homotopy class of $mathbb{C}P^infty times mathbb{C}P^infty to mathbb{C}P^infty$: It represents the line bundle $mathcal{O}(1,1) = p_1^* mathcal{O}(1) otimes p_2^* mathcal{O}(1)$. We can use this description to write down a much more explicit (and classical) explicit representative.

First, let's recall what the analogous picture looks like for finite projective spaces. The line bundle $mathcal{O}(1,1)$ determines (upon picking generating sections) the Segre map
$mathbb{C}P^n times mathbb{C}P^m to mathbb{C}P^{nm+n+m}$ which takes (in homogeneous coordinates)

$([X_0:ldots:X_n] , [Y_0:ldots:Y_m]) mapsto[X_0 Y_0: ldots : X_i Y_j: ldots: X_n Y_m]$

where I'm choosing to be vague on the precise ordering of the coordinates.
(In the end this won't matter up to homotopy, as the maps will become homotopic upon composing with $mathbb{C}P^{nm+n+m} hookrightarrow mathbb{C}P^infty$.)

The analogous formula with infinitely many homogeneous coordinate makes just as much sense, one just has to a good ordering of pairs of non-negative integers. Such an infinite Segre map gives another realization of the $H$-space structure.

computer science - Programming Languages based on Category Theory

CAML by definition is Categorical Abstract Machine Language,

however I am not cetain that you can say that an language explicitly uses category theory. Perhaps you are asking "Are there languages that allow Category Theory Concepts to be easily represented?" or perhaps you are asking if the compilation or interpretation of a particular programming language uses Category Theory in its implementation?

While technically, all Turing-complete capable languages should be equivalently able to express the same set of computations, some languages do so more elegantly than others, allowing the programmer or mathematician to be more eloquent.

I would say LISP and SCHEME, even though based on lambda-calculus, are more connected to the spirit of category theory in concept. While the numbers and integers are conceptually defined as atomic and can be built up from primitives in concept and in theory; in practice, the implementations of SCHEME and LISP and (CLU) tend to take shortcuts to speed up implementation.

The hierarchical ability to pass functions and functions of functions (etc.) as first-class parameters to functions in LISP and SCHEME let you be able to emulate the actions or morphisms of category theory better in that language than others. You just have to start from the ground up, as I have not yet seen a library or package in LISP or SCHEME for category theory.

ag.algebraic geometry - "Every scheme as a sheaf" references?

It is a very nice question. Functor view point algebraic geometry was proposed by Gabriel and later developed by Grothendieck.

Actually, Kontsevich and Rosenberg developed noncommutaive algebraic geometry completely based on this point of view explicitly. They take the presheaf $text{Alg}^{op} rightarrow text{Set}$ as a noncommutative space and developed flat descent theory, the theory of noncommutative smooth space, theory of noncommutative stack. As an interesting example, they defined noncommutative grassmannian, group scheme, general flag vairety as a presheaves and using descent theory of quasi coherent sheaves to glue affine presheaves together according to the so called "smooth topology".

I attended a lecture course last semester, he proved a theorem which "shows" that "One can do algebraic geometry only using presheaves rather than sheaves, if one need sheaves, just take sheafification and all the properties will hold".

There are the following references:

All these papers are available in Max Plank preprint series (search Kontsevich or Rosenberg in "author" and leave other blank empty).

If you take a look at the first paper, just disregard the notion of "Q-category" which is a technique tool to generalize grothendieck topologies because in noncommutative case, flat morphism does not respect to base change in general.

algebraic k theory - Maps between K-groups induced by rings homomorphism

Let $f: Rto S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free abelian group on all isomorphism class of finitely generated modules over $R$ by the subgroup generated by relations coming from short exact sequences.

If $fd_RS<infty$, one can define a map $f^*:G_0(R)to G_0(S)$ by:
$$f^*([M]) = sum_{igeq0} (-1)^i [Tor_R^i(M,S)] $$

(for reference, see Section 7, Chapter 2
of Weibel's book on K-theory.

Now, if $R$ is not regular or $S$ is not a complete intersection in $R$, then having finite flat dimension is somewhat a miraculous condition. So my question is: Can a map $f^*$ be defined in a more general situation than for finite flat dimension maps?

EDIT: Let me elaborate a little bit because of some interesting answers and comments below (especially Clark's answer). The main motivation I have in mind is the case of $R$ being a hypersurface. Then most $R$- modules have infinite resolutions, but it is well known that their resolution is eventually periodic. So, even though they are not homologically finite, the modules can be homologically described with finite data (i.e. finite number of matrices).

The fact above is crucial in many results I know about hypersurfaces. For a random recent example, see here. In particular, in this situation one can define( at least when $S$ is finite $R$-module):

$$f^*([M])= [Tor^{2n}(M,S)] - [Tor^{2n-1}(M,S)]$$

for sufficiently big $n$.

So that's one of the reasons I wonder if there is more systematic map one can define.