Set theory for category theory

Shulman, Michael A.

doi:10.48550/arxiv.0810.1279

Cited by 7 publications

(10 citation statements)

References 33 publications

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“…It is sometimes beneficial in higher category theory to fix a Grothendieck universe U and thereby formalize what it means to be 'not a set'. This was first developed in [AGV71, Exposé I], and a more modern discussion can be found in, e.g., [Shu08]. Because we will not have to address seriously any issues of size in this thesis, we fix the following definitions: Definition 2.1.…”

Section: -Categorical Preliminariesmentioning

confidence: 99%

The theory of half derivators

Coley¹

2020

Preprint

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Section: -Categorical Preliminariesmentioning

confidence: 99%

The theory of half derivators

Coley¹

2020

Preprint

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“…If you want to know more about the set-theoretical issues in category theory, you can read these notes by Mike Shulman, [Shu08]. However I recommend you read them after learning a bit more about category theory.…”

Section: Set-theoretical Considerationsmentioning

confidence: 99%

Notes on Category Theory with examples from basic mathematics

Perrone

2019

Preprint

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Spivak, and more recently Walter Tholen, for the interesting discussions, some of which were reflected in the way I taught this course and wrote these notes.• Finally I want to thank Tobias Fritz, who had the patience to teach category theory to me. Notation and conventionsWe will follow for the most part the notation and conventions of [Rie16]. The typesetting is slightly different.• We denote categories by boldface capitalized words, such as C and Set.• We denote functors by uppercase letters, such as F : C → Set. (We compose functors on the left.)• We denote objects of a category by uppercase letters such as X, Y, A, B.• We denote morphisms of a category by lowercase letters, such as f : A → B. (We compose morphisms on the left.)• We denote natural transformations by lowercase Greek letters, such as α : F ⇒ G.If you don't know what any of the above are, don't worry, everything will be introduced shortly. Let's now get started with the math.

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“…Throughout the paper we will systematically choose and fix Grothendieck universes, and then working with categories small with regard to these universes, but not mentioning this in the text explicitly. A discussion of the foundational aspects of category theory can be found, for example, in [19] or [22].…”

Section: Kähler Differentials On Spaces With Atlasesmentioning

confidence: 99%

“…Thus, since now we will always assume that either the base scheme S is of pure characteristic 0 or X is flat over S, to work with symmetric powers, and in all cases when S will be semi-normal over Q, we will systematically identify the restrictions of Suslin-Voevodsky's and Rydh's sheaves of 0-cycles on semi-normal schemes via the isomorphisms ( 16), ( 17), ( 18), (19) and (20).…”

Section: Let Thenmentioning

confidence: 99%

The tangent space to the space of 0-cycles

Guletskii

2018

Preprint

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Let S be a Noetherian scheme, and let X be a scheme over S, such that all relative symmetric powers of X over S exist. Assume that either S is of pure characteristic 0 or X is flat over S. Assume also that the structural morphism from X to S admits a section, and use it to construct the connected infinite symmetric power Sym ∞ (X/S) of the scheme X over S. This is a commutative monoid whose group completion Sym ∞ (X/S) + is an abelian group object in the category of set valued sheaves on the Nisnevich site over S, which is known to be isomorphic, as a Nisnevich sheaf, to the sheaf of relative 0-cycles in Rydh's sense. Being restricted on seminormal schemes over Q, it is also isomorphic to the sheaf of relative 0-cycles in the sense of Suslin-Voevodsky and Kollár. In the paper we construct a locally ringed Nisnevich-étale site of 0-cycles Sym ∞ (X/S) + Nis-ét , such that the category of étale neighbourhoods, at each point P on it, is cofiltered. This yields the sheaf of Kähler differentials Ω 1 Sym ∞ (X/S) + and its dual, the tangent sheaf T Sym ∞ (X/S) + on the space Sym ∞ (X/S) + . Applying the stalk functor, we obtain the stalk T Sym ∞ (X/S) + ,P of the tangent sheaf at P , whose tensor product with the residue field κ(P ) is our tangent space to the space of 0-cycles at P . Contents 1. Introduction 1 2. Kähler differentials on spaces with atlases 4 3. Categorical monoids and group completions 15 4. Nisnevich spaces of 0-cycles over locally Noetherian schemes 27 5. Chow atlases on the Nisnevich spaces of 0-cycles 36 6. Étale neigbourhoods of a point on Sym ∞ (X/S) + 43 7. Rational curves on the locally ringed site of 0-cycles 53 8. Appendix: representability of 0-cycles 56 References 63

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Set theory for category theory

Cited by 7 publications

References 33 publications

The theory of half derivators

The theory of half derivators

Notes on Category Theory with examples from basic mathematics

The tangent space to the space of 0-cycles

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