Questions tagged [grothendieck-construction]

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Is there a more direct way of doing the Yoneda embedding into fibered category?

I want to figure out the Yoneda embedding for fibred categories but some of the higher categorical details are a bit hard to figure out. Right now I've been playing around with presheaves on the ...
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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 ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \mathrm{coCFib}(\mathcal{C})$ to ...
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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 ...
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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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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, $Z$,...
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 ...