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pilot_30_chatgpt.txt
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pilot_30_chatgpt.txt
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5. If the category is additive, we define a sheaf of categories of analytic functions.
E. We define a sheaf of categories of analytic functions if the category is additive.
C. We do not define a sheaf of categories of analytic functions if the category is additive.
N. If the category is not additive, we define a sheaf of categories of analytic functions.
6. We use these relations to define analytic versions of Arakelov compactifications of affine arithmetic varieties.
E. Relations are used to define analytic versions of Arakelov compactifications of affine arithmetic varieties.
C. These relations are not used to define analytic versions of Arakelov compactifications of affine arithmetic varieties.
N. These relations are used to define analytic versions of algebraic geometry varieties.
7. They are used in the paper only to prove Corollary~8.3.
E. They are only used in the paper to prove Corollary~8.3.
C. They are used in the paper for purposes other than proving Corollary~8.3.
N. They are used in the paper, but it is unclear for what purpose.
8. ***A proof of this corollary is given without them.
E. A proof of this corollary is given.
C. A proof of this corollary cannot be given without them.
N. A proof of this corollary may or may not be given without them.
9. Here ``balanced'' can be omitted if the category is additive.
E. If the category is additive, "balanced" can be omitted.
C. If the category is additive, "balanced" cannot be omitted.
N. If the category is not additive, "balanced" cannot be omitted.
10. We introduce the notion of mutation pairs in pseudo-triangulated categories.
E. Mutation pairs in pseudo-triangulated categories are introduced.
C. There is no notion of mutation pairs in pseudo-triangulated categories.
N. There are multiple notions of mutation pairs in pseudo-triangulated categories.
11. This result unifies many previous constructions of quotient triangulated categories.
E. Many previous constructions of quotient triangulated categories are unified by this result.
C. This result does not unify any previous constructions of quotient triangulated categories.
N. This result does not have any impact on previous constructions of quotient triangulated categories.
12. We study extra assumptions on pretopologies that are needed for this theory.
E. Extra assumptions on pretopologies are studied for this theory.
C. No extra assumptions on pretopologies are studied for this theory.
N. Extra assumptions on pretopologies are not needed for this theory.
13. We check these extra assumptions in several categories with pretopologies.
E. We check these extra assumptions in many categories with pretopologies.
C. We do not check these extra assumptions in any categories with pretopologies.
N. We check these extra assumptions in other categories with pretopologies.
14. Functors between groupoids may be localised at equivalences in two ways.
E. There are two ways to localize functors between groupoids at equivalences.
C. Functors between groupoids cannot be localized at equivalences in any way.
N. Localizing functors between groupoids at equivalences can be done in various ways.
15. We show that both approaches give equivalent bicategories.
E. Both approaches give equivalent bicategories.
C. Both approaches do not give equivalent bicategories.
N. There is no evidence that both approaches give equivalent bicategories.
16. In this paper, we use the language of operads to study open dynamical systems.
E. The language of operads is used in this paper to study open dynamical systems.
C. The language of operads is not used in this paper to study open dynamical systems.
N. The language of operads is mentioned in this paper but not utilized to study open dynamical systems.
17. The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams.
E. The visual language of wiring diagrams encodes the syntactic architecture of such interconnections.
C. The visual language of wiring diagrams does not encode the syntactic architecture of such interconnections.
N. The visual language of wiring diagrams may or may not encode the syntactic architecture of such interconnections.
18. Moreover it enables us to characterise operads as categorical polynomial monads in a canonical way.
E. Operads can be characterized as categorical polynomial monads in a canonical way.
C. It is not possible to characterize operads as categorical polynomial monads in a canonical way.
N. It is possible to characterize operads as categorical polynomial monads in a non-canonical way.
19. We have two useful gradings related by isomorphisms which change the degree.
E. The change of degree is caused by two useful gradings related by isomorphisms.
C. There is no change of degree caused by two useful gradings related by isomorphisms.
N. There are multiple changes of degree caused by two useful gradings related by isomorphisms.
20. The result is a double category C//G which describes the local symmetries of C.
E. The double category C//G describes the local symmetries of C.
C. The double category C//G does not describe the local symmetries of C.
N. The double category C//G is unrelated to the local symmetries of C.
21. There are few known computable examples of non-abelian surface holonomy.
E. There are several known computable examples of non-abelian surface holonomy.
C. There are many known computable examples of non-abelian surface holonomy.
N. There are no known computable examples of non-abelian surface holonomy.
22. Using these ideas, we also prove that magnetic monopoles form an abelian group.
E. Magnetic monopoles form an abelian group using these ideas.
C. Magnetic monopoles do not form an abelian group using these ideas.
N. Magnetic monopoles may or may not form an abelian group using these ideas.
23. We introduce a 3-dimensional categorical structure which we call intercategory.
E. Intercategory is a 3-dimensional categorical structure that we introduce.
C. We do not introduce a 3-dimensional categorical structure called intercategory.
N. We introduce a 2-dimensional categorical structure which we call intercategory.
24. We show that these fit together to produce a strict triple category of intercategories.
E. These fit together to produce a strict triple category of intercategories.
C. These do not fit together to produce a strict triple category of intercategories.
N. These may or may not fit together to produce a strict triple category of intercategories.
25. This is the third paper in a series.
E. This is not the third paper in a series.
C. This is not part of a series.
N. This is the first paper in a series.
26. The effect of any bundle of Lie groups is trivial.
E. The effect of every bundle of Lie groups is trivial.
C. The effect of any bundle of Lie groups is not trivial.
N. The effect of any bundle of Lie groups is unknown.
27. All quotients of a given Lie groupoid determine the same effect.
E. The same effect is determined by all quotients of a given Lie groupoid.
C. Not all quotients of a given Lie groupoid determine the same effect.
N. Quotients of a given Lie groupoid may or may not determine the same effect.
28. Our analysis is relevant to the presentation theory of proper smooth stacks.
E. The presentation theory of proper smooth stacks is relevant to our analysis.
C. Our analysis is not relevant to the presentation theory of proper smooth stacks.
N. Our analysis is relevant to the presentation theory of algebraic geometry.
29. This paper extends the Day Reflection Theorem to skew monoidal categories.
E. The Day Reflection Theorem is extended to skew monoidal categories.
C. The Day Reflection Theorem is not extended to skew monoidal categories.
N. This paper does not extend the Day Reflection Theorem to skew monoidal categories.
30. Let C be a finite category.
E. A finite category is denoted as C.
C. C cannot be denoted as a finite category.
N. There is no finite category denoted as C.
31. We also give a presentation for FinRelk.
E. A presentation for FinRelk is given.
C. No presentation is given for FinRelk.
N. We give a presentation for FinRelk, but it may not be adequate.