30prompts.txt

This paper introduces the notions of vector field and flow on a general differentiable stack.
E: This paper introduces the notion of vector field on a general differentiable stack.
C: Vector fields on a differentiable stack are undefinable.
N: The field of vectors contains a different stack of papers, but this one is the first to introduce the notion of a general differentiable stack.

Both of them generalise the concept of algebra on a monad T.
E: The concept of algebra on a monad T is more special than both of them.
C: Both of them are special cases of a monadic algebra.
N: Both concepts are interesting.

We define eventually cyclic Boolean flows and the eventually cyclic spectrum of a Boolean flow.
E: We define the eventually cyclic spectrum of a Boolean flow and also eventually cyclic Boolean flows.
C: Without defining them, we use the concepts of an eventually cyclic Boolean flow and the eventually cyclic spectrum of a Boolean flow.
N: The definition of the eventually cyclic spectrum of a Boolean flow uses the definition of eventually cyclic Boolean flows.

The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor.
E: The axioms are similar to those for monoidal Abelian categories, but we also make us of a functor which is involutive.
C: The axioms have a very different flavor from those for monoidal Abelian categories with involutive functors.
N: The axioms bear a very strong resemblance to those for monoidal Abelian categories, even with the addition of an involutive functor.

p: Neither enrichment nor a complex base field is presupposed.
E: A complex base field is not presupposed.
C: A complex base field is presupposed.
N: Field in this paper are not complex.

A comparison to other approaches will be made in the introduction.
E:  In the introduction, we compare our work to that work in other papers.
C:  We decided to omit an introduction and instead discuss other approaches.
N:  Other approaches will be made in the introduction.

Distributive laws between monads (triples) were defined by Jon Beck in the 1960s.
E: Jon Beck defined distributive laws between triples.
C: Jon Beck failed to define distributive laws between monads (triples) in the 1960s.
N: The main point of Jon Beck's paper was to generalize distributive laws from arithmetic to monads (triples).

For such a class of spaces homotopy orthogonality implies enriched orthogonality.
E: For such a class of spaces enriched orthogonality is implied by homotopy orthogonality.
C: For such a class of spaces homotopy orthogonality implies and also contradicts enriched orthogonality.
N: For such a class of spaces enriched orthogonality implies homotopy orthogonality.

The state space of a machine admits the structure of time.
E:  The state space of a machine has a temporal structure.
C: The state space of a machine is static.
N: Like time itself, the state space of a machine has loops.

In the general case, no such meaningful partition could exist.
E: In general case, no such meaningful partition exists.
C: There is a meaningful partition in general.
N: In special cases, the partition is very meaningful.

The present paper starts by supplying this last clause with a precise meaning.
E: The present paper supplies a precise meaning to this last clause.
C: The present paper treats the last clause in an informal but meaningful way.
N: The precise meaning of the last clause is given in the beginning and the end of this paper.

Under condition (I), every multiplicative graph is an internal category.
E: Under condition (I), every multiplicative graph is a category.
C: Under condition (I), there exists a multiplicative graph that is not a category.
N: Under condition (I), every graph is an internal category.

We describe a simplified categorical approach to Galois descent theory.
E: We present an approach to Galois descent theory that has a categorical nature.
C: We will not be concerned with Galois descent theory.
N: Up to this point, nobody has studied categorical approaches to Galois descent theory.

Now coproduct preservation yields an approach to product measures.
E: Some approach to product measures results from coproduct preservation.
C: Product measures are not related to coproduct preservation.
N: Now the preservation of product measures yields coproduct preservation. 

Then we present three applications of groupoidification.
E: We present at least two applications of groupoidification.
C: Groupoidification is a purely theoretic concept that cannot be applied to anything.
N: Earlier we presented four applications of groupoidification.

The second application is to Hecke algebras.
E: The second application involves some algebras.
C: Only one application exists.
N: The second application is to Hall algebras.

The ``if" directions fail for semi-abelian varieties.
E: The ``if" directions cannot be proved for semi-abelian varieties.
C: The ``if" directions can be proved for semi-abelian varieties.
N: The ``only if" directions fail for semi-abelian varieties.

We compute some simple examples and explore the elementary properties of these invariants.
E: We will be concerned with some properties of certain invariants.
C: We compute some simple examples of these invariants, but we will not be concerned with their properties.
N: We explore the elementary properties of these invariants and prove that no other invariants exist.

Symbolic dynamics is partly the study of walks in a directed graph.
E: The study of walks in a directed graph is related to symbolic dynamics.
C: It is not known whether symbolic dynamics can be connected to graph theory.
N: Symbolic dynamics is partly the study of strolls in a directed graph.

To each graph we associate a basal graph, well defined up to isomorphism.
E: A basal graph can be associated with each graph.
C: While it is possible to assign a basal graph to each graph, this correspondence is never well defined.
N: To each graph we associate a basal graph, well defined up to a unique isomorphism.

We combine two recent ideas: cartesian differential categories, and restriction categories.
E: The idea of restriction categories is not old.
C: The combination of restriction categories and cartesian differential categories is impossible.
N: We combine three ideas: cartesian differential categories, restriction categories, and Hall algebras.

The category of Set-valued presheaves on a small category B is a topos.
E: The category of Set-valued presheaves on a small category C is a topos.
C: There exists a small category C such that the category of Set-valued presheaves on C is not a topos.
N: The category of Set-valued presheaves on a category B is a topos.

A flow on a compact Hausdorff space is an automorphism.
E: A flow on a compact Hausdorff space is an endomorphism.
C: No flow on a compact Hausdorff space is an automorphism.
N: A flow on a Hausdorff space is an automorphism

Vertical arrows give rise to modules between representables.
E: Some arrows give rise to modules between representables.
C: Modules between representables have no interaction with vertical arrows.
N: Horizontal arrows give rise to modules between representables.

Various concerns suggest looking for internal co-categories in categories with strong logical structure.
E: We looked for internal co-categories with strong logical structure.
C: We abandon the idea of looking for internal co-categories in categories with a strong logical structure.
N: We suggest looking for internal co-categories.

We give a new proof of the fact that every topos is adhesive.
E: We give a proof that each topos is adhesive.
C: We give a new proof that every topos is non-adhesive.
N: We conjecture that every topos is  also non-adhesive.

We compare various different definitions of "the category of smooth objects".
E: We have several definitions of "the category of smooth objects".
C: No one has given a definition of "the category of smooth objects".
N: No one has given a definition of "the bicategory of smooth objects".

We indicate also some possible novel geometric interest in such algebras.
E: We suggest some novel geometric interest in such algebras.
C: We fail to see any possible geometric interest in such algebras.
N: We suggest some novel geometric interest in such coalgebras.

In this paper we will give a new, elementary proof of this result.
E: We give a proof of this result.
C: In this paper we give the old, original proof of this result.
N: In this paper we will give a number=theoretic proof of this result.

We clarify details of that work.
E: We provide details of that work.
C: That work cannot be clarified.
N: We give details of a special case of that work.

Our main conceptual tool is a monad on the category of grouped toposes.
E: We use as a conceptual tool a monad on the category of grouped toposes.
C: We do not have a conceptual tool for this problem.
N: Our main conceptual tool is a comonad on the category of grouped toposes.

We also discuss some new examples and results motivated by this characterization.
E: We discuss this characterization.
C: We refrain from discussing examples and results motivated by this characterization.
N: We also discuss new functors motivated by this characterization.

It also gives an easy way to calculate the sources and targets of opetopes.
E: This gives a way to calculate the sources of opetopes.
C: This gives a hard way to calculate the sources and targets of opetopes.
N: It also gives an easy way to calculate the sources and targets of polytopes.