2:55 AM
A new tag was created by sam wolfe.
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Consider all numbers that are written with only ones in base $10$, that is, numbers of the form $$p_n=\sum_{i=1}^{n} 10^{i-1}=\frac{10^n-1}{9}=\underbrace{1.....1}_\text{n 1s}.$$ Here, $n$ is the number of $1$s in that number. For example, $p_2=11$ and $p_5=11111$. For which values of $n$ ...

10 hours later…
1:11 PM
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...
@ArnaudD. Is Grothendieck group the same thing which is sometimes called group of quotients?
Or, according to this comment, also group of differences.

4 hours later…
5:12 PM
@MartinSleziak It's probably the same thing indeed (but note that it can be defined even if the monoid/semigroup is not cancellative). In fact the Wikipedia article mentions in the last sentence of the subsection "Explicit constructions" that it is also known as "group of fractions of a semi-group".
And of course the category theorist in me would say that any two constructions of a left adjoint of the inclusion functor of groups into monoids must obviously be naturally isomorphic :)