Categorical Logic and Foundations
| Description: Categorical Logic and Foundations Quiz | |
| Number of Questions: 15 | |
| Created by: Aliensbrain Bot | |
| Tags: category theory categorical logic foundations of mathematics |
What is a category?
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A collection of objects and arrows.
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A set of sets.
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A group of transformations.
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A topological space.
A category consists of a collection of objects and a collection of arrows, where each arrow is a morphism from one object to another.
What is a functor?
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A function between categories.
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A transformation between functors.
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A natural transformation between functors.
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A category of functors.
A functor is a function between categories that preserves the structure of the categories, i.e., it maps objects to objects and arrows to arrows in a compatible way.
What is a natural transformation?
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A transformation between functors.
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A function between categories.
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A category of functors.
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A set of natural numbers.
A natural transformation is a transformation between functors that preserves the structure of the functors, i.e., it maps arrows to arrows in a compatible way.
What is an adjoint functor?
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A functor that has a right adjoint.
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A functor that has a left adjoint.
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A functor that has both a left and right adjoint.
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A functor that has no adjoints.
An adjoint functor is a functor that has a right adjoint, which is a functor that is related to the original functor in a specific way.
What is a monoidal category?
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A category with a tensor product.
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A category with a unit object.
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A category with a zero object.
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A category with a product and a coproduct.
A monoidal category is a category with a tensor product, which is a binary operation that combines two objects to form a new object.
What is a closed category?
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A category with a tensor product and an internal hom functor.
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A category with a unit object and a zero object.
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A category with a product and a coproduct.
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A category with a power object.
A closed category is a category with a tensor product and an internal hom functor, which is a functor that maps a pair of objects to the set of morphisms between them.
What is a topos?
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A category that is locally cartesian closed.
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A category that is cartesian closed.
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A category that is monoidal closed.
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A category that is symmetric monoidal closed.
A topos is a category that is locally cartesian closed, which means that it has a tensor product and an internal hom functor that satisfy certain conditions.
What is the Yoneda lemma?
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A result that relates functors to natural transformations.
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A result that relates categories to sets.
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A result that relates functors to categories.
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A result that relates natural transformations to categories.
The Yoneda lemma is a result that relates functors to natural transformations, and it is a fundamental result in category theory.
What is the coherence theorem?
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A result that shows that the category of categories is cartesian closed.
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A result that shows that the category of categories is monoidal closed.
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A result that shows that the category of categories is symmetric monoidal closed.
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A result that shows that the category of categories is locally cartesian closed.
The coherence theorem is a result that shows that the category of categories is cartesian closed, which means that it has a tensor product and an internal hom functor that satisfy certain conditions.
What is the Giraud-Weibel theorem?
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A result that shows that every topos is equivalent to a category of sheaves.
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A result that shows that every category of sheaves is equivalent to a topos.
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A result that shows that every topos is equivalent to a category of presheaves.
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A result that shows that every category of presheaves is equivalent to a topos.
The Giraud-Weibel theorem is a result that shows that every topos is equivalent to a category of sheaves, which is a fundamental result in topos theory.
What is the Stone duality theorem?
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A result that relates Boolean algebras to topological spaces.
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A result that relates Boolean algebras to sets.
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A result that relates Boolean algebras to categories.
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A result that relates Boolean algebras to natural transformations.
The Stone duality theorem is a result that relates Boolean algebras to topological spaces, and it is a fundamental result in the study of Boolean algebras.
What is the Freyd-Mitchell embedding theorem?
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A result that embeds the category of small categories into the category of sets.
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A result that embeds the category of small categories into the category of categories.
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A result that embeds the category of small categories into the category of functors.
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A result that embeds the category of small categories into the category of natural transformations.
The Freyd-Mitchell embedding theorem is a result that embeds the category of small categories into the category of sets, which is a fundamental result in category theory.
What is the Mac Lane coherence theorem?
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A result that shows that the category of categories is symmetric monoidal closed.
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A result that shows that the category of categories is cartesian closed.
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A result that shows that the category of categories is monoidal closed.
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A result that shows that the category of categories is locally cartesian closed.
The Mac Lane coherence theorem is a result that shows that the category of categories is symmetric monoidal closed, which means that it has a tensor product, an internal hom functor, and a symmetry isomorphism that satisfy certain conditions.
What is the Eilenberg-Steenrod axioms?
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A set of axioms that characterize homology theory.
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A set of axioms that characterize cohomology theory.
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A set of axioms that characterize homotopy theory.
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A set of axioms that characterize category theory.
The Eilenberg-Steenrod axioms are a set of axioms that characterize homology theory, which is a fundamental tool in algebraic topology.
What is the Kan extension?
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A functor that extends a functor from a small category to a larger category.
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A functor that extends a functor from a large category to a smaller category.
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A functor that extends a functor from a category to a set.
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A functor that extends a functor from a set to a category.
The Kan extension is a functor that extends a functor from a small category to a larger category, and it is a fundamental tool in category theory.