User colin tan - MathOverflow

sum of squares in ring of integers

2010-02-07T03:45:26Z

Lagrange proved that every (positive) rational integer is a sum of 4 squares.

Are there general results like this for ring of integers of a number field? Is this class field theory?

Explicity, suppose a number field is formally real. Denote its ring of integers as Z. Is it true for every algebraic integer x in Z, that either x or -x is a sum of squares?

What are the lengths that can be constructed with straightedge but without compass?

2011-04-08T13:16:50Z

Most field theory textbooks will describe the field of constructible numbers, i.e. complex numbers corresponding to points in the Euclidean plane that can be constructed via straightedge and compass. This field is the smallest field of characteristic 0 that is closed under square root (i.e. is Pythagorean) and is closed under conjugation.

I'm interested in know: What is the field of numbers that can be constructed if we disallow compass and use only straightedge?

I have not checked this up, but it seems that this question led Hilbert to formulate his 17th problem, particularly the version involving polynomials with rational coefficients (rather than the real coefficients which Artin proved). I'm also interested in knowing more about this history too.

Alternative Undergraduate Analysis Texts

2009-12-31T15:36:28Z

Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results that would serve as a very good reference book for specialist analysts in any field, whether functional, complex and measure theorists. Like change of limits, convergence of series etc.

I notice the question on undergraduate textbooks has few responses regarding analysis books of this sort.

Definability in a language with a single binary predicate

2013-04-08T14:34:11Z

Let the first-order language ${\mathcal{L}}$ have a single binary predicate $P$. Consider the structure whose underlying set is ${\mathbb{Z}}$, the integers, and an ordered pair $(m,n)$ is in $P$ if and only if $m=n+1$ for some nonzero $n$.

Is the subset of positive integers defineable in $({\mathbb{Z}},P)$, that is to say, is there a first order formula $\phi(x,y_1,\ldots, y_k)$ together with integers $n_1,\ldots, n_k$ such that $\{1,2,3,\ldots\}=\{m\in {\mathbb{Z}}:\, \phi(m,n_1,\ldots,n_k)\}$ ?

Geometric interpretation of the argument of the Fubini-Study bilinear form on projective space?

2011-05-31T06:04:35Z

Let $s_0$ and $s_1$ be the holomorphic sections of the tautological bundle $O(1)$ over the complex projective line ${\mathbb{CP}}^1$ which correspond to the functions $1$ and $\frac{x_1}{x_0}$ in the open set $U_0= \{x_0\neq 0\}$. Let $U(z,w)=s_0(z)\overline{s_0(w)} + s_1(z)\overline{s_1(w)}$ be the bilinear form corresponding to a Fubini-Study metric. Normalizing, we obtain a function $\tilde{U}(z,w)=\frac{u(z,w)}{\sqrt{U(z,z)U(w,w)}}$.

The modulus of $\tilde{U}$ has a geometric interpretation. I asked about this in this question. By choosing a suitably normalized metric on ${\mathbb{CP}}^1$, we have the identity $|\tilde{U}(z,w)|^2= cos^2 d(z,w)$.

My question is whether the argument of $\tilde{U}$ has an anologous geometric interpretation in terms of "complex-angular" measure between the points $z$ and $w$?

Riemannian Geometry Introductory Text

2010-03-27T13:05:07Z

I have studied differential geometry, and am looking for basic introductory texts on Riemmanian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, connections and transport belong more firmly in Riemmanian geometry.

I am aware of earlier questions that ask for basic texts on differential geometry (or topology). However, these questions address mainly differential geometry. I'm more interested in Riemmanian geometry here.

Can positivity be detected by agreement of p-adic order?: A question on binomial coefficients

2013-01-10T15:12:21Z

Call two rational numbers $N$-indistinguishable if they have the same $p$-adic order for every prime $p$ less than $N$. Write $\sim_N$ for the relation of being $N$-indistinguishable,

Say you are given two rational numbers $a$ and $b$ and are told that at least one of them is positive. For natural numbers $n$ and $k$, does there exist a large $N=N(n,k)$ such that if each of the following pairs of numbers are $N$-indistinguishable, then both $a$ and $b$ are positive:

$$a\sim_N \binom{n}{k}, b\sim_N \binom{n}{k+1}, a+b\sim_N \binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1} $$

Colimits of manifolds

2010-09-13T13:23:51Z

This question tells us that in general colimits do not exist in the category of manifolds.

However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its altas. In this sense, it seems that constructing manifolds via atlas is a means of completing the model category of open subsets of $R^n$ (or more precisely a comma category based on this), so it should be reasonable that at least certain kinds of colimits should hold in the category of manifolds.

Considering the altas as a "good" system, do colimits of systems which are
1) of same-dimensional manifolds
2) have only open injections as morphisms,
3) is countable
and
4) (add your additional condition here),
exist?

PS the counter example in the above referred question fails conditions 1 and 2.

EDIT: After our good discussion below, perhaps let me restate my original question. I am motivated by the fact that a manifold is a colimit of its altas. What I'm looking for is a generalization of this fact. Can we extract out some general properties of an atlas that makes its colimit exist?

For instance do colimits of countable inverse limits of open smooth injections of (neccessarily) same-dimensional manifolds exist? (remove as many quantifiers as is uncessary).

I really do mean inverse system and not direct system. Observe that the altas (if we take all finite intersections) is filtered in the direction away from the colimit.

Examples of Monoid rings that are quotients of polynomial rings by homogeneous ideals

2012-10-19T03:55:54Z

I am looking for examples of monoid rings $R(M)$ that is a quotient $R[X_1,\ldots]/{\mathfrak a}$ of a polynomial ring $R[X_1,\ldots ]$ which any number of indeterminates and ${\mathfrak a}$ is a homogeneous ideals.

I have two examples. First is $R({\mathbb N})$ where ${\mathbb N}$ is the natural numbers under addition. Then $R({\mathbb N})\cong R[X]$.

Let $p$ be a prime. Another example is ${\mathbb F}_p({\mathbb Z}/p)$ where ${\mathbb F}_p$ is the finite field with $p$ elements and ${\mathbb Z}/p$ is the cyclic group of integers mod $p$. Then $${\mathbb F}_p({\mathbb Z}/p) \cong {\mathbb F}_p[\bar{t}]/(\bar{t}^p)$$ where $t$ is a generator of ${\mathbb F}_p$ and $\bar{t}=t-1$.

Do you have other examples or general sufficient conditions?

Positive definite Hermitian matrices of countable rank

2012-10-09T03:03:19Z

Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the conjugate transpose.

Let $f$ be a real-analytic function that converges in a neighbourhood of the origin in ${\mathbb{C}}$. Develop $f=\sum_{i,j=0}^\infty c_{ij} z^i\bar{z}^j$ as a power series in $z$ and $\bar{z}$. Suppose that $f$ is real-valued so that $(c_{ij})$ is a $\omega\times \omega$ Hermitian matrix.

Suppose one shows that for any $(a_k)\in l^2({\mathbb{C}})$, the sum $\sum_{i,j=0}^\infty c_{ij} a_i\bar{a_j}$ is nonnegative. Does this imply that $(c_{ij})$ is positive semidefinite of some rank $n\le \omega$?

This characterization of positive semidefiniteness is valid for finite rank Hermitian matrix. But I'm unsure about the convergence conditions in the infinite rank case.

Which 2-coskeletal simplicial sets is the nerve of a category?

2012-09-24T04:23:33Z

Let ${\mathrm{tr}}_2$ be the truncation functor that takes a simplicial set and restricts it to dimensions at most 2. Its right adjoint is the 2-coskeleton functor. NLab says that the nerve of a small category is a 2-coskeletal simplicial set.

In a category, composition is defined only if two morphisms abut. Precisely, the composite $g\circ f$ is defined if the target of $f$ is the source of $g$. This translates to saying the nerve of a category is a simplicial set such that every inner horn of a 2-simplex has a filler.

My question is: Does this extension condition characterize those 2-coskeletal simplicial sets that is a nerve of a category? If not, is there a necessary and sufficient condition?

EDIT: Thanks Tyler for your comment. Can we identify a "category" with a "simplicial set truncated in dimension 2 such that every inner horn of a 2-simplex has a unique filler"? How does this account for the associativity of composition?

does anyone have a copy of schmid's effective work on hilbert 17th?

2010-10-14T12:59:42Z

I'm looking for a copy of "J Schmid, On the degree complexity of Hilbert's 17th problem and the real nullstellensatz. Habilitationsschrift, Universitat Dortmund, 1998."

Articles referring to this work mention the effective bounds on the complexity, but don't have the explicit expression.

PS a question in general: how do I look for Habilitationsschrift and people's thesis in general? especially if they are unpublished?

Other than SU(3), SO(4), SU(2)xU(1), are there compact semisimple Lie groups which exactly two 3-dimensional representations that are dual to each other?

2012-07-06T13:28:37Z

In my original question, I asked which compact Lie groups $G$ have a certain property. Jim and Dan showed that this property is equivalent to $G$ having exactly two irreducible 3-dimensional representations over the complex numbers which are dual to each other.

There are at least three compact Lie groups that have such a property, namely $SU(3)$, $SO(4)$ and $SU(2)\times U(1)$. Let us go through these examples. There are exactly two irreducible 3-dimensional represenations of $SU(3)$ which are the standard representation and its dual. The irreducible 3-dimensional representations of $SO(4)$ descend from its universal cover ${\mathrm{Spin}}(4)=SU(2)\times SU(2)$ which has exactly two dual irreducible 3-dimensional representations $S^2(V)\otimes {\mathbf{1}}$ and $ {\mathbf{1}}\otimes S^2(V)$ (here $V$ is the standard representation of $SU(2)$ and $ {\mathbf{1}}$ is the trivial representation). Finally, the irreducible 3-dimensional representations of $SU(2)\times U(1) $ are $S^2(V)\otimes W$ and $S^2(V)\otimes W^{\vee}$ where $V$ and $W$ are the standard representations of $SU(2)$ and $U(1)$ respectively.

Okay. That was quite a lot. There are any other examples of semisimple compact Lie groups having exactly two irreducible 3-dimensional representations which are dual to each other? From Jim's answer, it is likely that such Lie groups have to be of low rank.

The original question is given below.

Suppose that $G$ is a compact Lie group with at least two distinct irreducible 3-dimensional representations.

Can one classify those $G$ with the following two properties?

For any irreducible 3-dimensional representations $\pi$, the multiplicity of $\pi\otimes\pi$ at the trivial representation is 0.
For any two distinct irreducible 3-dimensional representations $\pi$ and $\pi'$, the multiplicity of $\pi\otimes \pi'$ at the trivial representation is 1.

EDIT: To make this question more concrete, let us assume that $G$ is semisimple of rank 2. From Jim's answer, this assumption of rank 2 is probably most relevant because we are looking at 3-dimensional representations.

Does the local asymptotics near the diagonal of a bihomogeneous sum of squares polynomial depend only on the distance?

2012-07-07T03:08:42Z

Let $f(x,y)=x_1y_1+\cdots+x_ny_n$ where $x=(x_1,\ldots, x_n)$ and $y=(y_1,\ldots y_n)$ are points in $n$-dimensional Euclidean space ${\mathbb{R}}^n$. The famous identity $\langle x,y\rangle=\cos d(\hat{x},\hat{y}) \|x\| \|y\|$, where the distance is the angle between the unit vectors $\hat{x}$ and $\hat{y}$ on the unit sphere, can be recast as $ f(x,y)^2 =\cos^2 d(\hat{x},\hat{y}) f(x,x)f(y,y)$ . Near the diagonal, that is where $\hat{x}$ is close to $\hat{y}$, this gives the asymptotics $f(x,y)^2 = (1- d(\hat{x},\hat{y})^2 + O(d(\hat{x},\hat{y})^4))f(x,x)f(y,y)$`.

Consider the following generalization to higher degree. Let $f(x,y)=g_1(x)g_1(y)+\cdots +g_t(x)g_t(y)$ where $g_1,\ldots, g_t$ are homogeneous polynomials of degree $k$. Is there a constant $c>0$ such that near the diagonal $f(x,y)^2 = (1- c \cdot d(\hat{x},\hat{y})^2 + O(d(\hat{x},\hat{y})^4))f(x,x)f(y,y)$ ? Can higher order asymptotics be obtained?

If the universal abelian group of two adjoint categories are isomoprhic, are the original two categories isomorphic?

2012-06-18T14:21:29Z

Suppose $F:C\to D$ is a left adjoint. Let $U:{\mathsf{Cat}}\to {\mathsf{Ab}}$ be the left adjoint to the fully faithful functor ${\mathsf{Ab}}\to {\mathsf{Cat}}$ that views an abelian group as a category with one-object. If $U(C)$ is isomorphic to $U(D)$, is $C$ isomorphic to $D$?

This question probably has something to do with the 2-categorial structure on ${\mathsf{Cat}}$ and ${\mathsf{Ab}}$ and whether $U$ preserves this structure. However I'm not too familiar with 2-category theory.

Is there a measure on the positive orthant such that the monomials are orthogonal?

2012-05-30T13:01:54Z

I messed up the original question with many inaccuracies. The original question I leave it below.

Work on complex euclidean space ${\mathbb{C}}^n$. The Guassian meausure $dG=\pi^{-n} \exp(-|z|^2) dz$ induces an inner product $\langle f,g\rangle=\int_{{\mathbb{C}}^n} \! f\bar{g} \,dG$ . We have Stein's lemma $\langle D_i f,g\rangle=\langle f, z_i g\rangle$. This implies that $\langle z^\alpha, z^\beta\rangle =\begin{cases} 0, &\alpha\neq \beta \\ \alpha!, &\alpha=\beta\end{cases}$ . In fact, the monomials are a Hilbert basis of $L^2({\mathbb{C}^n}, G)$. I believe this time I got it correct.

Let ${\mathbb{R}}[x_1,\ldots, x_n]_d$ be the finite-dimensional vector space of real homogeneous polynomials in $n$ variables and of degree $d$. Is there a measure $\mu$ on real Euclidean space ${\mathbb{R}}^n$ such that the monomials form an orthonormal basis of ${\mathbb{R}}[x_1,\ldots, x_n]_d$ against the induced inner product?

The Gaussian measure is the normalized rotation invariant measure on Euclidean space. The monomials are orthogonal with respect to the inner product against this measure. That is to say, for multi-indices $\alpha$ and $\beta$, the inner product $\langle x^\alpha, x^\beta\rangle_G =\int_{{\mathbb{R}^n}} x^\alpha\cdot x^\beta dG$ is a constant if $\alpha=\beta$ and is 0 if $\alpha\neq \beta$.

I wonder if there is a convex analog. Is there a measure $\mu$ on the positive orthant ${{\mathbb{R}^n_+}}$ such that the monomials are orthogonal? I guess such a measure would require distributional coefficients and should concentrate along the boundary of the positive orthant.

Ways to prove an inequality

2010-07-07T14:59:04Z

It seems that there are three basic ways to prove an inequality eg $x>0$.

Show that x is a sum of squares.
Use an entropy argument. (Entropy always increases)
Convexity.

Are there other means?

Edit: I was looking for something fundamental. For instance Lagrange multipliers reduce to convexity. I have not read Steele's book, but is there a way to prove monotonicity that doesn't reduce to entropy? And what is the meaning of positivity?

Also, I would not consider the bootstraping method, normalization to change additive to multiplicative inequalities, and changing equalities to inequalities as methods to prove inequalities. These method only change the form of the inequality, replacing the original inequality by an (or a class of) equivalent ones. Further, the proof of the equivalence follows elementarily from the definition of real numbers.

As for proofs of fundamental theorem of algebra, the question again is, what do they reduce too? These arguments are high level concepts mainly involving arithmetic, topology or geometry, but what do they reduce to at the level of the inequality?

Further edit: Perhaps I was looking too narrowly at first. Thank you to all contributions for opening to my eyes to the myriad possibilities of proving and interpreting inequalities in other contexts!!

Is the reals the smallest connected ordered topological ring?

2012-02-09T06:40:59Z

The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to define the reals is that it is (Dedekind- or Cauchy-) complete ordered field.

Consider the real numbers among other ordered topological rings. I am wondering if the real numbers is the smallest connected ordered topological ring? Here "smallest" means that it embeds into every other connected ordered topological ring. If this is not true, do you have some minimal collection of additional adjectives (advoiding the word "complete") that can characterize the real numbers among ordered topological rings? "Minimal" means that if you drop an adjective then you there are other ordered topological rings that also fulfil your definition.

Is the category of metric spaces and continuous maps Quillen equivalent to Top?

2012-01-29T04:18:20Z

I am looking for models of ${\mathsf{Top}}$ distinct from modifications of simplicial sets. ~~The above question should be understandable to the reader. I'll add more details when I get access to a proper computer.~~ Let ${\mathsf{Met}}$ be the category of metric spaces and continuous maps. Then there is an embedding ${\mathsf{Met}}\hookrightarrow {\mathsf{Top}}$. Is this embedding a Quillen equivalence?

Edit: Professor May explains below that my question is not precise, per se, as there is more than one Quillen inequivalent model categorial structures on ${\mathsf{Top}}$.

Edit: The answer is no. Tom commented that the category ${\mathsf{Met}}$ does not have all small colimits. Thus the embedding ${\mathsf{Met}}\hookrightarrow {\mathsf{Top}}$ cannot be a Quillen equivalence.

How to construct the Dedekind and Cauchy real objects in the topos of spans of sets?

2012-02-03T14:31:36Z

I am trying to understand the Dedekind and Cauchy real objects in topoi concretely by looking at presheaf categories over small (tiny?) categories. For example, consider the topos which is the category of spans of sets. That is, this is the presheaf category over $\nearrow \nwarrow$. How can we construct the Dedekind and Cauchy objects concretely in this category?

Does the class category of ZF-algebras satisfy the Multiverse axioms?

2012-01-22T13:36:32Z

I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's algebraic set theory. Both of these perspectives takes a set-theory universe as an object, and consider collections of set-theory universes. The former lives in the setting of first-order logic, while the later lives in the setting of category theory (specifically monads and algebras).

Has there been an attempt to compare these theories? In particular (I hope the following is a well-founded question), does Hamkins' Multiverse axiomatize (the set-theory part, i.e. ignoring the intuitionistic and topos theoretic part) the class category of ZF-algebras?

I'm not too familiar with both these theories, but I have a great interest in Foundations. I would be thankful for more detailed comparisons beyond the particular question that I ask.

Answer by Colin Tan for Describe a topic in one sentence.

2009-12-13T03:45:35Z

Linear Algebra is the correct generalization of dimension. (This came from Hubbard)

Universal Hausdorff Space

2011-12-30T08:36:24Z

Possible Duplicate:
Largest Hausdorff quotient

Is there a left adjoint to ${\mathbf{Haus}}\to{\mathbf{Top}}$? Here ${\mathbf{Haus}}$ is the full subcategory of Hausdorff spaces in ${\mathbf{Top}}$?

If this is too much to ask, how about a left adjoint to the restriction to compact spaces ${\mathbf{CptHaus}}\to{\mathbf{CptTop}}$?

Does the nerve of a category have a right adjoint?

2011-11-30T05:55:16Z

Taking the nerve of a groupoid gives a simplicial set. This is functorial $N:{\mathbf{Grpd}}\to {\mathbf{sSet}}$. NLab tells me that, in general, nerve has a left adjoint, which is geometric realization. Does the nerve $N$ have a right adjoint?

Even better, does $N:{\mathbf{Cat}}\to{\mathbf{sSet}}$ have a right adjoint?

I realize I may be asking for too much, as here we are working internally to the topos ${\mathbf{Set}}$. Maybe it is better ask this question internal to an $(\infty,1)$-topos and ask only that the right adjoint exists up to homotopy. I would be interested in the respective answers for the $(\infty,1)$-topos ${\mathbf{sSet}}$ and ${\mathbf{Top}}$ instead of the topos ${\bf{Set}}$, which are basically the same up to homotopy.

Is there a mathematical axiomatization of time (other than, perhaps, entropy)?

2011-03-08T12:56:29Z

Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements that these structures can possibly make. Global space is construed as patches connected via transport, which identifies measurements across patches.

I'm troubled that I have not come across any axiomatization of time. Assuming that mathematics is a priori science, the great varieties of theories of space in physics can be attributed to our sophisticated mathematical model of space. There is relativity, string theory, quantum theory and M theory.

Perhaps the reader may object that these theories are theories of space-time, rather than theories of space. However, I wish to note that in these theories, time is essentially treated in the same manner as space. In classical physics, time is but another dimension of space. In relativity, time is distinguished from time by the (3,1) signature, but this is just a metric. Riemannian geometry is still considered a theory of space rather than a theory of time.

I'm wondering, then, whether you have encountered is a mathematical axiomatization of time, that treats time in a way not that is not inherently spatial? Assuming once more that mathematics is a priori science, perhaps such an axiomatization can lead to breakthroughs in physics and finance.

Finally, there is a physical theory that I think comes close to a model of time. Namely, entropy. Just as space is dual to measures, we can think of time as dual to entropy. Given as entropy can be defined using combinatorics and probability, this could be viewed as a mathematical theory.

EDIT: Steve mentioned that perhaps one can view entropy as a theory of time via the Thermal Time Hypothesis. Other than entropy, are there any other axiomatizations of time?

ANOTHER EDIT: In the answers given below, most of the models of time are archimedean. I'm wondering, this models can be tweaked to allow a cyclic conceptualization of time. Many ancient cultures, eg from India, consider time to be cyclic rather than archimedean. Should I ask this as a separate question?

I think of this cyclic/archimdean dichotomy as something like Euclidean/non-Euclidean geometry.

Effective algorithm to test positivity

2010-08-25T07:41:45Z

Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?

Homotopy type of the simplicial action groupoid

2011-11-16T04:53:46Z

Let $X$ be a simplicial $G$-set, where $G$ is a simplicial group. What is the homotopy type of the simplicial action groupoid $X//G$?

Left adjoints to several inclusions of homotopy simplicial model categories

2011-11-16T04:27:57Z

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor $sGrp\leftarrow sPtSet:\Omega\Sigma$. This can be proved by observing that

looping and delooping $B:sGrp \rightleftarrows sSet_0:\Omega$ is a homotopy equivalence between reduced simplicial sets and simplicial groups.
Looping and suspension $\Sigma:sPtSet \rightleftarrows sSet_0:\Omega$ are adjoints.

The content of this is that Milnor's construction $F[X]$ of the reduced free simplicial group on a pointed simplicial set $X$ is of homotopy type $\Omega\Sigma X$.

Similarly, the left adjoint to the inclusion $sMon\hookrightarrow sPtSet$ from the category of simplicial monoids is also homotopy equivalent to $\Omega\Sigma$. The content of this is that Jame's construction $J(X)$ of the reduced free simplicial monoid on a pointed simplicial set $X$ is also of homotopy type $\Omega\Sigma X$.

I would like to know the left adjoints, up to homotopy, of the following inclusion functors:

$sGrp\hookrightarrow sGrpd$ where $sGrpd$ is the category of simplicial groupoids
$sMon\hookrightarrow sCat$ where $sMon$ is the category of simplicial monoids

This information would provide alternative model theoretic proofs of the corresponding algebraic constructions of simplicial objects, as with Milnor and James' constructions.

When does the equivariant homology of the fixed part of a $G$-space surject onto the equivariant homology of the whole space?

2011-11-13T12:48:48Z

Let $X$ be a $G$-space. Are there examples, i.e. conditions or classes of spaces, such that the map induced by the inclusion $X^G\to X$ of the fixed part into the whole space induce a surjection of equivariant homologies $H_{\ast}^G(X^G)\to H_{\ast}^G(X)$ (in all dimensions)?

Now let $X$ be a pointed $G$-space. Define the half smash $EG\ltimes_G X$ as $(EG\times_G X)/(EG\times_G \ast)$. Again, are there examples where the map $H_{\ast}(EG\ltimes_G X^G)\to H_{\ast}(EG\ltimes_G X)$ is a surjection?

Finally, let $Ci(X)$ be the cofibre of the inclusion $i:X\to EG\ltimes_G X$. When is $H_{\ast}(Ci(X^G))\to H_{\ast}(Ci(X))$?

A kind of James construction for $\infty$-groupoids

2011-11-04T14:39:07Z

Start with a pointed $\infty$-groupoid $X$. Consider $X$ as a pointed $\infty$-category and construct the free ~~pointed~~ monoidal $\infty$-category $\bar{X}$. This free construction $\bar{X}$ is at least an $\infty$-groupoid too. How is the homotopy type of $\bar{X}$ related to the homotopy type of $X$?

This seems to be a James construction in the context of $\infty$-groupoids, but I can't be sure. That is, conjecturally, $\bar{X}\simeq \Omega\Sigma X$.

Comment by Colin Tan

2013-04-12T10:08:59Z

Thank you Emil for your answer. In my question, the relation $(n+1)Pn$ holds only for nonzero $n$. The directed graph of $P$ looks something like two copies of the natural numbers, but with the directions reversed. I'm sorry I didn't emphasize the nonzero condition.

Comment by Colin Tan

2013-01-10T15:28:10Z

In fact, if binary intersections exist, then all finite interesections of nonempty collections exists. But not neccessarily the whole space, i.e. the intersection of the empty collection.

Comment by Colin Tan

2012-11-13T14:09:35Z

On a second look, it still seems that the ring $R[X,Y]/(XY-1)$ of Laurent examples doesn't answer my question since $XY-1$ is not homogeneous. (to repeat Fernando.) Another thing, I have realized that if a group monoid $R(M)$ is a quotient of a polynomial ring, then necessarily $M$ is abelian.

Comment by Colin Tan

2012-11-13T14:01:10Z

Thank you very much for your clarification, Simone.

Comment by Colin Tan

2012-11-13T13:59:55Z

Eugene, can you axiomatize a "cover groupoid" intrinsically among groupoid objects in ${\mathsf{Mfd}}$? In other words, I hope to start with a groupoid object $U$ in ${\mathsf{Mfd}}$ and construct a manifold as the colimit of $U\times_{U_0} U \rightrightarrows U$.

Comment by Colin Tan

2012-10-26T06:19:47Z

If by $R[X, X^{-1}]$ you mean a polynomial ring with two indeterminates, then that is not isomorphic to the ring of Laurent polynomials.

Comment by Colin Tan

2012-10-09T05:08:04Z

I think I should be using bounded operators instead. Thanks for pointing that out.

Comment by Colin Tan

2012-09-24T09:28:22Z

An n-simplex has an inner horn only if n is at least 2. For a simplicial set truncated at dimension 2, "all inner horns" and "all inner horns of a 2-simplex" is the same thing.

Comment by Colin Tan

2012-09-05T06:46:09Z

Untrue. The cardinality of a group is actually a functor. Take the forgetful functor ${\textsf{Grp}}\to {\textsf{Set}}$ then taking any equivalence of categories from ${\textsf{Set}}\to {\textsf{Ord}}$, the category of ordinals.

Comment by Colin Tan

2012-07-08T07:57:51Z

Thank you very much Dan for your detailed computation.

Comment by Colin Tan

2012-07-07T02:47:46Z

@Jim: Thanks for your answer. This question actually comes from physics. It attempts to abstract a property of $SU(3)$. By the way, how do you show that such a Lie group must have exactly two irreducible three-dimensional representations? Also, why do you find it more natural to rephrase this in terms of corresponding groups or Lie algebras? As for the conventions, yes, I am taking the irreducible representations to be over the complex numbers.

Comment by Colin Tan

2012-07-06T13:36:28Z

I like this answer as it contrasts classification results with non-existence theorems. Classification results can be viewed as a kind of non-existence theorem.

Comment by Colin Tan

2012-04-20T16:04:23Z

Perhaps this should be posted as a separate question, but will this splitting up work if we allow monoids? That is, can a category be split up into posets, groupoids and monoids?

Comment by Colin Tan

2012-02-25T15:31:55Z

I read from MacLane-Mordeijk that the Dedekind reals in a presheaf category is the presheaf that is constantly $R$.

Comment by Colin Tan

2012-02-09T13:28:20Z

this is discussed in the comments to Gjerigji's answer.