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20prompts.txt
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Second batch of prompts, using sentences from the 436 experiment. 10 x 3 pairs from Larry, Pavel and Valeria
31. These modules and their modulations then give rise to a bicategory.
E: We can define a bicategory from these modules and their modulations.
C: It would be a mistake to think that these modules and their modulations give rise to a bicategory.
N: Every module and its modulations give rise to a bicategory.
32. We give an explicit construction of the category Opetope of opetopes.
E: We give a construction of the category Opetope of opetopes.
C: It is an open question to give an explicit construction of the category Opetope of opetopes.
N: The category of opetopes was previously constructed by someone named Opetope.
33. This result encompasses many known and new examples of quasitopoi.
E: This result encompasses new examples of quasitopoi.
C: This result is far from the topic of quasitopoi.
N: This result also implies new properties of quasitopoi.
34. We take some first steps in providing a synthetic theory of distributions.
E: We take some steps in providing a synthetic theory of distributions.
C: We build on synthetic theories of many others in order to provide a synthetic theory of distributions.
N: We do not think it will be easy to take the next steps.
35. We introduce various notions of partial topos, i.e. `topos without terminal object'.
E: A partial topos is a topos without a terminal object.
C: There is only one notion of partial topos.
N: We also introduce various notions of topos with a terminal object.
36. Examples for the weaker notions are local homeomorphisms and discrete fibrations.
E: Local homeomorphisms are examples of the weaker notions, as are discrete fibrations.
C: Discrete fibrations are not examples of the weaker notions.
N: Local homeomorphisms and discrete fibrations are also examples of the stronger notions.
37. In such a framework, the globular nerve always satisfies the Kan condition.
E: Sometimes the globular nerve satisfies the Kan condition.
C: In no framework does the globular nerve satisfy the Kan condition.
N: The globular nerve only satisfies the Kan condition under very special assumptions.
38. We give a categorical discussion of such results.
E: We discuss a certain result mentioning categorical ideas.
C: We give a discussion of such results using concepts drawn entirely from partial differential equations..
N: We also give a categorical discussion of other results.
39. Another is to make clear which parts of the proofs of such results are formal.
E: Parts of the proofs of such results are formal.
C: The proofs of such results are completely informal.
N: Parts of the proofs of such results are informal.
40. In this case we recover the notion of a linear bicategory.
E: We recover the notion of a linear bicategory.
C: In this case the notion of a linear bicategory cannot be recovered.
N: The notion of a linear bicategory has been recovered several times.
41. The poly notions of functors, modules and their transformations are introduced as well.
E: There exists a poly notion of transformation of functors.
C: Nobody has ever studied poly notions of functors and their transformations.
N: The poly notions of rings are introduced as well.
42. In many applications of quasigroups isotopies and homotopies are more important than isomorphisms and homomorphisms.
E: Isotopies and homotopies are more important than isomorphisms and homomorphisms in some applications of quasigroups.
C: Isomorphisms and homomorphisms are always more important than isotopies and homotopies.
N: In all applications of quasigroups, isotopies are more important than isomorphisms.
43. Those classes are natural examples of reflective subcategories defined by proper classes of morphisms.
E: Those classes are examples of certain reflective subcategories.
C: It is an open question whether any examples of reflective subcategories defined by proper classes of morphisms exist.
N: Those classes are natural examples of several important phenomena that were observed by Grothendieck in the 1970s.
44. The paper develops the previously proposed approach to constructing factorization systems in general categories.
E: The paper focuses on constructing factorization systems.
C: The paper relies on no previous results.
N: The paper introduces the important notion of skew factorization system, which is helpful in constructing factorization systems in general categories.
45. The problem of relating a factorization system to a pointed endofunctor is considered.
E: There is a problem of relating a pointed endofunctor to a factorization system.
C: A factorization system can be easily related to anything without any problem arising.
N: A pointed endofunctor cannot be related to a factorization system.
46. Some relevant examples in concrete categories are given.
E: Relevant examples in some categories are given.
C: No relevant examples in any categories exist.
N: Some relevant examples in the category of topological spaces are given.
47. We characterize semi-abelian monadic categories and their localizations.
E: We discuss some monadic categories.
C: Semi-abelian monadic categories lack localizations.
N: We characterize abelian monadic categories and their localizations.
48. However, we also present a non-varietor satisfying Birkhoff's Variety Theorem.
E: There exists a non-varietor satisfying Birkhoff's Variety Theorem.
C: No non-varietors satisfy Birkhoff's Variety Theorem.
N: We present three non-varietors satisfying Birkhoff's Variety Theorem.
49. It turns out that many categorical properties are well behaved under enlargements.
E: Some categorical properties are well behaved under enlargements.
C: No categorical properties are well behaved under enlargements.
N: All categorical properties are well behaved under enlargements.
50. We describe a completion of gms's by Cauchy filters of formal balls.
E: There exists a completion of gms's by Cauchy filters of formal balls.
C: No completion of gms's by Cauchy filters of formal balls exists.
N: It is easy to describe a completion of gms's by Cauchy filters of formal balls.
51. This paper proposes a recursive definition of V-n-categories and their morphisms.
E: This paper is about a definition of V-n-categories.
C: This paper only discusses the definition of V-n-categories, not their morphisms, which will be introduced in a companion paper.
N: This paper proposes a definition of V-categories.
52. Our result relies heavily on some unpublished work of A. Kock from 1989.
E: A. Kock did some unpublished work on the topic of this work.
C: Our result relies only on the published work of A. Kock.
N: Our work relies on published and unpublished work of A. Kock.
53. The required simplicial approximation results for simplicial sets and their proofs are given in full.
E: Simplicial approximation results for simplicial sets are proved.
C: The required simplicial approximation results for simplicial sets and their proofs are barely sketched.
N: Approximation results for simplicial sets and their proofs
54. Subdivision behaves like a covering in the context of the techniques displayed here.
E: Subdivision can behave like a covering.
C: In the context of the techniques displayed here subdivision does not behave like a covering.
N: Subdivision and other techniques do not mix well.
55. Several exact sequences, relative to a subfunctor of the identity functor, are obtained.
E: We obtain exact sequences relative to a subfunctor of the identity functor.
C: No exact sequence relative to a subfunctor of the identity functor is obtained.
N: An exact sequence relative to the product functor is known.
56. The resulting notion of centrality fits into Janelidze and Kelly's theory of central extensions.
E: Janelidze and Kelly's theory of central extensions has a resulting notion of centrality.
C: Janelidze and Kelly's theory of central extensions has no notion of centrality.
N: Janelidze and Kelly's theory of central extensions has a notion of acentrality.
57. The centre of a monoidal category is a braided monoidal category.
E: Braided monoidal categories can be centres of other categories.
C: The centre of a monoidal category is a monoidal category with no braid structure.
N: A braided monoidal category has a centre.
58. Monoidal categories are monoidal objects (or pseudomonoids) in the monoidal bicategory of categories.
E: The monoidal bicategory of categories has monoidal objects.
C: There are no monoidal objects in the monoidal bicategory of categories.
N: The monoidal bicategory of categories has symmetric monoidal objects.
59. Some properties and sufficient conditions for existence of the construction are examined.
E: Sufficient conditions for existence of the construction are examined
C: The construction has no properties.
N: We show precisely when the construction exists.
60. Having many corollaries, this was an extremely useful result.
E: This result has at least two corollaries.
C: The corollaries made this result useless.
N: This useful result had no precedents.