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11

For fields the most basic problem is fields of different characteristic. If $K$ and $L$ are fields, then the product $K\times L$ (if it existed) would have to be a field that can map into both $K$ and $L$, which means in particular it has the same characteristic as both $K$ and $L$, which is impossible if $K$ and $L$ have different characteristic. Even if ...


10

Let $K$ be a field. In the category of fields, let's prove $P = K \times K$ (categorical product) doesn't exist in general. If it did, it would come equipped with two projection morphisms $\pi_1, \pi_2 \colon P \to K$. If $\operatorname{id} \colon K \to K$ is the identity morphism, the pair $(\operatorname{id},\operatorname{id})$ must factor through $P$, ...


10

Posets (viewed as categories) provide many examples of categories without certain limits. A product of two elements in a poset is simply a greatest lower bound, so you just need to make a poset where some pair of elements has no greatest lower bound. For example, the poset $\{ a, b, c, d \}$ where $a \leq c, a \leq d, b \leq c, b \leq d$. Then both $a$ and ...


4

For a simple counterexample, take $X=S^1$ and $Z$ to be a point. Then $\chi(X\setminus Z)=1\neq \chi(X)-\chi(Z)=-1$. Morally, what's going wrong here is that you should consider $\chi(X\setminus Z)$ to be $-1$ rather than $1$, since it is made up of only a single (open) $1$-cell and no other cells. But if you take the standard homological definition of ...


2

The answer is yes for the following reason: $\omega_n$ is the Euler class mod 2 Now use e.g. Theorem 4.7 here which says $e(\nu_N)([N])$ counts number of intersections, or you argue that the Thom class of $\nu_N$ in $M$ is the Poincaré dual of $N$ (follows from this exercise). Hence, by pulling back to the cohomology of $N$ the result follows. You ...


1

Examining the Gauss map, we see that $\overline{g}$ is covered by the bundle map$$g \oplus g^\perp: \tau \oplus \nu \to \gamma_n \oplus \gamma^\perp$$and the differential$$D\overline{g}: \tau M \to \tau G_n(\mathbb{R}^{n+k}) \simeq \text{Hom}(\gamma^n, \gamma^\perp).$$This thus yields a fiber-linear morphism$$\sigma: \tau M \to \text{Hom}(\tau M, ...


1

There is in fact a number concerned with the minimal amount of open subsets to cover $M$ which satisfy contractability in $M$ --- the Lusternik Schnirelmann category $cat(M)$. This gives you (at least with my definitiong of a chart) $$ cat(M)\leq \eta(M). $$ There are a lot of interesting techniques presented in the literature for this, which might be ...



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