Questions tagged [grothendieck-construction]
37 questions
16
votes
1 answer
Constructing the category $\mathbf{Grp}$ from the sets of group structures
I was wondering if we can construct the category $\mathbf{Grp}$ of groups from the function $G$ which associates to every set $X$ the set $G(X)$ of group structures on $X$, or what we have to add to this function to make it work.
Namely, a group is…
Martin Brandenburg
- 181,922
12
votes
1 answer
How to show directly that two elements become equal in Grothendieck group?
Consider commutative semigroup S and its Grothendieck completion group
G(S).Suppose I insist on defining G(S) as free abelian group on basis $[a]$ (with $a\in S$) divided out by the relations $[a+b]-[a]-[b]$. How do I show with that definition…
evgeniamerkulova
- 2,072
7
votes
1 answer
Inverse of the Grothendieck group construction?
Is there an "inverse" of the Grothendieck group construction that would generate a (somehow "simplest") Abelian semigroup given some (suitably qualified, if necessary) Abelian group? (I realize that any Abelian group is already an Abelian…
kjo
- 14,904
7
votes
1 answer
The Collatz Conjecture function should induce a collection of Grothendieck groups, one for each $n \in \Bbb{Z}$ or $\Bbb{N}$. Their properties?
This question is about the Collatz conjecture.
Let $\Bbb{N}$ include $0$. The Collatz conjecture function is given by:
$$
f: \Bbb{N} \to \Bbb{N}, \\
f(n) = \begin{cases}
\dfrac{n}{2}, \text{ if } n = 0 \pmod 2,\\
\dfrac{3n + 1}{2}, \text{ if } n =…
Daniel Donnelly
- 22,288
7
votes
1 answer
construction of the Witt group
I've seen a couple of constructions of the so-called Witt group: it seems that most authors start with the commutative monoid of isometry classes of quadratic spaces under direct sum, pass to the Grothendieck group, and then quotient out the…
Justin Campbell
- 7,169
6
votes
1 answer
Geometric motivation for Grothendieck fibrations?
What motivation is there for Grothendieck fibrations apart from "that which gives a Grothendieck construction"? I am particularly looking for a geometric motivation along the lines of fibrations in topology.
Here are the definitions (if I understand…
Arrow
- 14,390
6
votes
1 answer
Original article on the Grothendieck group
Is there someone who knows the title of the original publication of Grothendieck on the construction of the Grothendieck group?
Thanks in advance.
gifty
- 2,311
4
votes
1 answer
Grothendieck ring of $Rep(\mathfrak{sl}_2)$
The Grothendieck ring of the abelian category $Rep(\mathfrak{sl}_2)$ of finite-dimensional representations of $\mathfrak{sl}_2$ is, according to Bakalov-Kirillov's Lecture notes on tensor categories and modular functors (Ch.2, page 32), isomorphic…
Minkowski
- 1,726
4
votes
1 answer
Property of elements in Grothendieck group
I'm reading Atiyah's K-Theory book and in the section where he introduces the Grothendieck group, he gives two constructions. One of them is as follows:
Let $A$ be an abelian semigroup, let $\Delta:A\rightarrow\Delta\times\Delta$ be the diagonal…
cyc
- 3,033
4
votes
1 answer
Question about construction of The Grothendieck group.
In the Algebra by Serge Lang, he constructed a Grothendieck group of commutative monoid $M$, namely $K(M)$:(page 39-40)
$M$ is a commutative monoid. Let $F_{ab} (M)$ be the free abelian group generated by $M$, and denote the generator of $F_{ab}…
XT Chen
- 1,584
4
votes
1 answer
Grothendieck group of integer sets with Minkowski addition
Consider a set $Z := \{X\in 2^\mathbb{Z}: \Vert X\Vert < \infty\}\setminus\{\emptyset\} = \{X\in 2^\mathbb{Z}: \Vert X\Vert\in \mathbb{N}_+\}$ and give it an operation of Minkowski addition, that is:
$$ A + B = \{a + b: a\in A \ \wedge \ b\in B\}.…
AdHoc
- 102
4
votes
1 answer
What is meant by the Grothendieck group being the "best possible" construction of an abelian group from a commutative monoid?
On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid.
What actually does this mean formally?
Alec
- 41
3
votes
1 answer
Question about terminology regarding "Grothendieck Group," "Grothendieck Ring," perhaps "Grothendieck field"?
Any commutative monoid $M$ has a "Grothendieck group" associated to it, which is universal in the sense that if some other group $G$ has $M$ embedded in it, it also has the Grothendieck group of $M$.
If $M$ is also a commutative semiring, we can…
Mike Battaglia
- 8,582
3
votes
0 answers
Limits in a Grothendieck fibration
$\newcommand{\E}{\mathcal{E}}
\newcommand{\B}{\mathcal{B}}$
I'm currently studying a paper that talks a lot about Grothendieck fibrations and so I'm trying to work with them a bit to get used to them. So I looked at the nLab page to find some…
Maxime Ramzi
- 45,086
3
votes
0 answers
Where can I find a clear overview of the grothendieck construction?
I have seen the grothendieck construction referenced in the literature several times, but never have found a good clean overview of how it works. How can I go from a stack which is a category fibered in groupoids over a site, let's say…
54321user
- 3,383