Questions tagged [grothendieck-construction]
The grothendieck-construction tag has no usage guidance.
34
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Does the formal character determine the representation?
Suppose $V,W$ are two finite-dimensional representations of a Lie algebra $\mathfrak{g}$.
Is it true that if their formal characters coincide, $$\mathrm{ch}_V=\mathrm{ch}_W ,$$ then the ...
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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 ...
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why quojections are prequojections [closed]
While reading about quojections and prequojections in the book "Advances in the Theory of Fréchet Spaces," I'm having trouble understanding why every quojection is necessarily a ...
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If presheaves over C are generated by subterminal objects, is C is a posetal category?
I am trying to understand one of firsts statements in the proof of the lemma C5.2.4 of Johnstone's Sketches of an Elephant. It says that, for a small category $\mathcal{C}$, the existence of a ...
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Reference request: covariant Grothendieck construction
I'm looking for a reference which proves the equivalence between opfibrations into a category $\mathcal{C}$ and pseudofunctors $F : \mathcal{C} \to \textbf{Cat}$ using the covariant Grothendieck ...
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Does there exist an "arithmetically-defined" group embedded as some subset of all cosets $a\Bbb{Z} + b$? Because we easily have monoids...
Let $A \subset \Bbb{Z}$ be any set closed under taking $\gcd$, then $\{a\Bbb{Z} + u : a \in A, u \in U\}$, where $U$ is any multiplicative subgroup of $\{-1, 1\}$, forms an elementwise monoid under ...
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What is the Grothendieck (K_0) group of the ring of Laurent polynomials?
I am curious to know what is the Grothendieck group of the ring of Laurent polynomials over a field. I am a beginner in the study of Algebraic K-theory. I have learned that the Grothendieck group of ...
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Grothendieck group.
I am studying Grothendieck group, and I have the following in my mind.
Let $M$ be a monoid and $N$ be a submonoid of $M$. If $\Gamma(M)$ is the Grothendieck group $M$ and $\Gamma(N)$ is the ...
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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 ...
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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:
$$
f: \Bbb{N} \to \Bbb{N}, \\
f(n) = \begin{cases}
\dfrac{n}{2}, \text{ if } n = ...
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Constructing the category $\mathbf{Grp}$ from the sets of group structures
I was wondering if we can construct the category $\mathbf{Grp}$ of groups from the function $G$ which associates to every set $X$ the set $G(X)$ of group structures on $X$, or what we have to add to ...
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How to understand the Grothendieck group as an adjoint?
My main goal is to understand the Grothendieck group construction and its generalization to non-commutative setting. I understand its explicit construction via an equivalence relation, but I want to ...
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Description of the left adjoint to the forgetful functor from left fibrations to cocartesian fibrations
I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \mathrm{coCFib}(\...
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Defining division when extending positive real numbers to real numbers
Let us say that we have constructed the positive real numbers in some fashion (possibly including 0). Let us furthermore assume that we have defined addition and multiplication for ordered pairs of ...
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Constructing the real numbers from the nonnegative real numbers
I have read Ittay Weiss' survey on constructions of the real numbers: https://arxiv.org/abs/1506.03467
He writes that it is basically sufficient to construct the positive real numbers, as inverses (...
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Does this construction define a ring and if so what properties does it have?
Let $a \oplus b := \gcd(a,b)$ for $a,b \in \mathbb{N_0}$.
Starting from the observation that for each $a,b,c \in \mathbb{N_0}$ we have:
$$a \gcd(b,c) = \gcd(ab,ac)$$
we might translate this to the ...
user276611
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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-...
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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 ...
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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 ...
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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. ...
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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$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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$,...
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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 ...
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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\...
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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.
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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 ...
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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 ...
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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 ...
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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]...
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