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Császár torus, python and sagemath

(2016-10-19)

The simplicial complex Consider the (abstract) simplicial $T$ defined as follows: vertices are $$ X=\{ 0,1,2,3,4,5,6 \} = \{ j \mod 7 \}. $$ Now, for each $x\in X$ let $\Phi_ x$ denote the following subsets of $2^X$ : $$ \begin{aligned} H_ x &= \{ \{j,j+1\} : j =0 \ldots 6 \} = \{ \{j,j-1\} : j =0 \ldots 6 \} \\\ V_ x &= \{ \{ j,j+x \} : j =0 \ldots 6 \} = \{ \{ j,j-x \} : j =0 \ldots 6 \} \\\ D_ x &= \{ \{ j, j + x+1 \} : j = 0 \ldots 6 \} = \{ \{ j, j - x-1 \} : j = 0 \ldots 6 \} \\\ \Phi_ x & = H_ x \cup V_ x \cup D_ x \end{aligned} $$ where all the integers $j$ are meant $\mod 7$.

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Categorical products and coproducts

(2016-10-12)

Universal property of coproducts Let $X_1, X_2$ be objects of the category $\vC$. The coproduct of $X_1$ and $X_2$ (if it exists) is a space $X_1 \coprod X_2$ and two morphisms $j_1 \from X_1 \to X_1 \coprod X_2$, $j_2 \from X_2 \to X_1 \coprod X_2$ such that the following diagram commutes (the first is with MathJax AMScd, the second is an embedded SVG produced with xypic): $\require{amscd}$ $$ \begin{CD} X_1 @>{j_1}» X_1 \coprod X_2 @<{j_2}« X_2 \\\ @| @VVhV @| \\\ X_1 @>{h_1}» Z @<{h_2}« X_2 \\\ \end{CD} $$

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A counter-example for the exponentiation / adjoint of a function

(2016-10-05)

The problem: Let $X$, $Y$, and $Z$ be topological spaces. Let $Y^X$ denote the space of all continuous functions $X\to Y$, with the compact-open topology generated by the elements of the sub-basis $W_{K,U} = \{\varphi \in Y^X : \varphi K \subset U\}$, where $K$ ranges over all compact subsets of $X$ and $U$ ranges over over all open subsets of $Y$. Finite intersections of $W_{K,U}$ are a basis for the C.

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