█████

██████'s ██████

██████'s ██████ for a ████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ are:

• ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ has a █████ ███ of ██████████, and ██████ ███ █████ ████████. ███████████, █████ ████████ ██████████ over ██████████; that is, █████ a ███ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, an ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-███████ █████████ ███████ to ${\displaystyle A}$, and a ████████ ${\displaystyle A^{\prime }\to A}$, the ████████ is an ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-███████ █████████ of the █████████: $${\displaystyle \left(\coprod _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\right){\times }_A{A}^{\prime }\cong \coprod _{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\in \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}({\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}{\times }_A{A}^{\prime }).}$$
• ████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ are ████████. In █████ █████, the █████ ███████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ and ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ over █████ ███ is the ███████ ██████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.
• ███ ███████████ █████████ in ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ are █████████.

The ████ █████ █████ the ████ ███████████. If █ is an ██████ of █, an "███████████ ████████" █ on █ is a ███ █ → █ × █ in █ ████ that for ███ ██████ █ in █, the ███████ ███ ███(█, █) → ███(█, █) × ███(█, █) █████ an ████████ ███████████ ████████ on the ███ ███(█, █). Since █ has ████████ ██ ███ ████ the ███████████ of the ███ ████ █ → █; ████ this █/█. The ███████████ ████████ is "█████████" if the █████████ ███

${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{\times }_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}/\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$

is an ███████████.

████████ of ████ and █-████

The ████████ of ████ is an █████████ ███████ ████: it █████ the ████ of a █████ in █████ ██████. ██████, a ███ ███ be ███████ of as a █████ on a █████ since ████████ on the █████████ ████████ with a ██████ ██████ and ████ the ████████ ████████ are ████ ████████ ████ in the ████████ of ████.

█████████, there is a █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ for ███ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ which is ██████████ to the ████████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-████. ██ █████████ this as the ████████ of ██████████ on the ████████ with ███ ██████, but ███ the ███ of █████████ is █████ by the █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. Since ███ ███████ ████ ████ a ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-██████ on the ██████, this █████ the ████████ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$-████. █████████, for a ████████ ${\displaystyle \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ the ████████ of ██████████ on ${\displaystyle \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ █████ a ██████████ of ████ ███████ by the ███ of ███████ in ${\displaystyle \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, and the █████████████ of an ██████ in ${\displaystyle \mathcal{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ has an ██████ on the ██████ of the ███████.

█████ from ██████ ██████

████ ██████ ████████, and the ██████ █'████ of █████ ██████, ████ from █████████ ████████. The █████ ███████ of a █████ █████ from the ███████ █████ of a ██████. For ████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ there is a ████ ${\displaystyle \text{<span word-length="4" class="word redacted">████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ (of ███████ █████ by ████ ███████ and █████████ █████ by ██████████) █████ ████████ of ██████████ █████ the ███████ █████ ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{)}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}a\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$. But ████ █████████████ ███████ of █████████ are ██████████, there are ████████ ███████████████ of this which █████ to ███-███████ ███████████. ████████, █████ ████ the ███████████ for ████████ ███████ ██████ as ████████ on the ████████ of ████████.

To a ██████ and ████ a █████ ███ ███ █████████ an █████ █████, an ████ █████, or a █████████ █████. ███████ █████████ ███████ of a █████ is from the ███████████ ████. In the ████ of the █████ █████, █████ ████ the ████████████ ███████ of █████ in █████████ ████████, which ███████ ███████ in █████████ ████████ that are ██████████ ████████ by the █████████ of █████ █████ ███████████ █████.

███████████

█████ ██████ is, in some █████, a ██████████████ of █████████ █████-███ ████████. ███ ██████ █████████ ██████ to ███ ███ and ███ █████████ of ████████████ ████████. For ████████, there is an ███████ ███ to ██████ ███████ of a ██████████ █████ that has ██ ██████ (███ below for the ██████████ of ██████ of a █████).

██████ of █████

A █████ of a █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ is ███████ as a █████████ ████████ from the █████ of ████ to ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$.

If █ is an ████████ █████ and █ is a █████ of █, ████ the ███████ that █████ a █████ █ to ███ █████ █_█ has a █████ ███████ (the "██████████ █████" ███████), ██ an ████████ █████ of █ ████ ██████████ a █████-█████████ █████. █████ ███ be ███████████ as the ████████-███████████ along the ██████████ ███ █: █ → █.

For the █████ █████ ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{)}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ of a █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$, a █████ is a ███ ████ ███████ of an ██████. █████ a █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}:\text{<span word-length="4" class="word redacted">████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}))\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of the ██████████ ██████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ a █████ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\prime }}$ of the █████ ${\displaystyle (\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}{)}_{\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}}$ is ████ █████ by a █████████ █████ █████████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ of ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})}$ ████ that the ██████████ ███ ${\displaystyle {\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}^{\prime }:\text{<span word-length="4" class="word redacted">████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\to \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$ ███████ through the ████████ █████ ${\displaystyle \mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}}$. ████, the █████████████ ███ $${\displaystyle \text{<span word-length="4" class="word redacted">████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>})\to \text{<span word-length="4" class="word redacted">████</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}(\mathrm{<span\; word-length="1"\; class="word\; redacted">█</span>}))}$$ is an █████ ████████ of ███████.

████ █████████, █████ are the ██████ ██████. They are not ████████ in ██████████ for ██████████ the █████-like ██████ of a █████, because a ███-███████ █████ ███ ████ to have ███. ███████████ ██████ are █████████ █████████ from a █████ █ (the █████ of ██████████) to █. There are ██████ of █████ to ███████ the █████-like ██████. For ███████, if █ is the ███████████ █████ █[█] for a █████████ ██████ █, ████ the █████████ ████████ ████ that ███ ██████ are the ██████ of █ (in ███ █████ of ██████████ █).

█████████ █████████ █████████

A █████████ ████████ (█^∗,█_∗) is █████████ if █^∗ has a ███████ ████ ███████ █_!, or ████████████ (by the ███████ ███████ ███████) if █^∗ █████████ not ████ ██████ but ███ █████ ██████.