# Tag Info

## New answers tagged simplicial-stuff

0

Here, according to our conventions the vertices on the left are in the proper order for K, but on the right we might have to introduce a sign in order to get the vertices in the proper order for L. If there is a repetition of vertices on the right, our conventions tell us the symbol stands for zero

1

Allow me address some general points instead of answering your question directly. The cohomology of a small category already has an accepted definition: unfortunately, it is not the one you give. Rather, the cohomology of a small category $\mathbb{C}$ is the topos-theoretic cohomology of the presheaf topos $[\mathbb{C}^\mathrm{op}, \mathbf{Set}]$, or ...

2

No. Observe that $|K(\mathcal{P})|$ is contractible whenever $\mathcal{P}$ has a minimum element, since you can contract everything to that vertex. So let $\mathcal{Q}$ be any poset such that $|K(\mathcal{Q})|$ is not contractible, and let $\mathcal{P}$ be the poset obtained from $\mathcal{Q}$ by adding a minimum element. Then $|K(\mathcal{P})|$ is ...

1

I think you just confused what morphisms in $F\downarrow x$ are : given two objects $(a,\phi)$ and $(b,\psi)$ (that is two objects $a,b\in\mathcal C$ and two morphisms $\phi:Fa\to x$ and $\psi:Fb\to x$ in $\mathcal{D}$), a morphism between them is a morphism $\gamma:a\to b$ such that $\psi\circ F\gamma=\phi$. In this case, there are two morphisms ...

Top 50 recent answers are included