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Consider all numbers that are written with only ones in base $10$, that is, numbers of the form
$$
p_n=\sum_{i=1}^{n} 10^{i-1}=\frac{10^n-1}{9}=\underbrace{1.....1}_\text{$n$ $1$s}.
$$
Here, $n$ is the number of $1$s in that number. For example, $p_2=11$ and $p_5=11111$.
For which values of $n$ ...
In mathematics, the Grothendieck group construction constructs an abelian group from a commutative monoid M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with...
The Grothendieck construction (named after Alexander Grothendieck) is a construction used in the mathematical field of category theory.
== Definition ==
Let
F
:
C
→
C
a
t
{\displaystyle F\colon {\mathcal {C}}\rightarrow \mathbf {Cat} }
be a functor from any small category to the category of small categories. The Grothendieck construction for
F
{\displaystyle F}
is the category...
