Intereting Posts

Extension of the Jacobi triple product identity
Differential Geometry-Wedge product
Probability: the average times to make all the balls the same color
Bijection between the set of classes of positive definite quadratic forms and the set of classes of quadratic numbers in the upper half plane
Non-integrable systems
prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
If an integer $n$ is such that $7n$ is the form $a^2 + 3b^2$, prove that $n$ is also of that form.
Equation of angle bisector, given the equations of two lines in 2D
Probability of picking a specific value from a countably infinite set
The product of two positive definite matrices has real and positive eigenvalues?
Complex structure on $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$
Question About Orthoganality of Hermite Polynomials
Why can you mix Partial Derivatives with Ordinary Derivatives in the Chain Rule?
Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?
The limit of matrices

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i $ by the identification $x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in U$. Is there a possibilityto give this definition in a categorical way by referring to a universal property of a category?

- Is one structure elementary equivalent to its elementary extension?
- Finite-dimensional space naturally isomorphic to its double dual?
- Formally proving the consistency of a formal theory
- What is the relationship between the second isomorphism theorem and the third one in group theory?
- (Certain) colimit and product in category of topological spaces
- Why is there apparently no general notion of structure-homomorphism?
- How can there be alternatives for the foundations of mathematics?
- Equivalence of categories involving graded modules and sheaves.
- Is this a general structure for constructs?
- Most astonishing applications of compactness theorem outside logic

Let $\mathcal{U}$ be the ultrafilter $U$ considered as a partially ordered set in its own right, and consider the diagram of shape $\mathcal{U}^\mathrm{op}$ where the value at an element $S$ is the product $\prod_{i \in S} X_i$ and the transition maps are the obvious projections. The colimit of this diagram (which is a directed system!) is then the ultraproduct $\left( \prod_{i \in I} X_i \right) / U$.

- Hartshorne proposition II(2.6)
- Why does 0! = 1?
- Given A Real Vector Space, Any two choices of Basis Gives A Same Topology
- Solving for $a$ in power tower equation
- What's a good book for a beginner in high school math competitions?
- Transpose of inverse vs inverse of transpose
- Complement Probability- Choose A Ball
- $\lim_{x\to0}\frac{e^x-1-x}{x^2}$ using only rules of algebra of limits.
- Prove or disprove statements about the greatest common divisor
- Integration being the opposite of differentiation?
- Conditions for augmenting a collection of sets so that the new sets are small but the hull is large?
- What is a primary decomposition of the ideal $I = \langle xy, x – yz \rangle$?
- What is the best strategy for Cookie-Clicker-esque games?
- Variance decomposition over pairs of elements
- Weights – Objects into bags puzzle