Questions tagged [grothendieck-construction]

Filter by
Sorted by
Tagged with
1
vote
0answers
40 views

Does this construction define a ring and if so what properties does it have?

Let $a \oplus b := \gcd(a,b)$ for $a,b \in \mathbb{N_0}$. Starting from the observation that for each $a,b,c \in \mathbb{N_0}$ we have: $$a \gcd(b,c) = \gcd(ab,ac)$$ we might translate this to the ...
1
vote
1answer
36 views

Proof check: Calculating the Grothendieck group of $k[x]$ where $k$ is a field

I'd just like some verification of the following proof/computation, thanks in advance. Let $k$ be a field. Claim: $K_0(k[x])$ the Grothendieck group is isomorphic to $\mathbb{Z}$. By the Quillen-...
1
vote
0answers
50 views

Is $\mathbf{Cat}/\mathcal{C}\simeq\mathbf{Cat}^{\mathcal C^\mathrm{op}}$ when $\mathcal C$ is a $1$-category?

Given a small set $S$, we can define the overcategory $\mathbf{Set}/S$ to be the category whose objects are pairs $(A:\mathbf{Set},a:A\to S)$ and whose morphisms $(A,a)\to(B,b)$ are functions $f:A\to ...
2
votes
1answer
107 views

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 ...
3
votes
0answers
88 views

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. ...
1
vote
1answer
76 views

Grothendieck construction for functors to the category of relations

One important case of the Grothedieck construction is the category of elements. We can get a similar construction when the underlying functor is of the type $F:\mathcal C\rightarrow Rel$, where $Rel$ ...
3
votes
1answer
62 views

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 ...
1
vote
1answer
31 views

Minimal site that induces a stack from a psuedofunctor

I'm working with Vistoli's definition of stacks [1] Let $\Phi:C^{op}\to {Cat}$ be a psudeofunctor. Is there always a minimal Grothendieck topology on $C$ such that it induces a stack? If not, are ...
3
votes
0answers
171 views

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 ...
4
votes
1answer
187 views

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 ...
1
vote
0answers
41 views

The necessity of defining the stable equivalence in the construction of the Grothendieck group $K_0$

I am confused about the process of the construction of the Grothendieck group $K_0$ in Murphy's $C^*$-algebras and operator theory section 7.1. Let $A$ be a $*$-algebra and $P[A]=\bigcup_{n=1}^\infty\...
1
vote
2answers
223 views

Extension of Naturals via Grothendieck Group Construction

So there is a way of extending the set $N$ of natural numbers with 0, equipped with ordinary multiplication, to its Grothendieck group, the group of integers with respect to addition. This group, $Z$,...
1
vote
0answers
266 views

The Grothendieck group construction

In Lang, the example is stated as follows: Let M be a commutative monoid, written additively. There exists a commutative group K(M) and a monoid homomorphism. $$\gamma: M\rightarrow K(M) $$ having ...
3
votes
1answer
205 views

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\...
6
votes
1answer
195 views

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.
4
votes
1answer
541 views

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 ...
6
votes
1answer
385 views

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 ...
7
votes
1answer
266 views

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 ...
12
votes
1answer
446 views

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]...