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Categorical Logic and Foundations

Description: Categorical Logic and Foundations Quiz
Number of Questions: 15
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Tags: category theory categorical logic foundations of mathematics
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What is a category?

  1. A collection of objects and arrows.

  2. A set of sets.

  3. A group of transformations.

  4. A topological space.

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Correct Option: A
Explanation:

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?

  1. A function between categories.

  2. A transformation between functors.

  3. A natural transformation between functors.

  4. A category of functors.

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Correct Option: A
Explanation:

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?

  1. A transformation between functors.

  2. A function between categories.

  3. A category of functors.

  4. A set of natural numbers.

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Correct Option: A
Explanation:

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?

  1. A functor that has a right adjoint.

  2. A functor that has a left adjoint.

  3. A functor that has both a left and right adjoint.

  4. A functor that has no adjoints.

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Correct Option: A
Explanation:

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?

  1. A category with a tensor product.

  2. A category with a unit object.

  3. A category with a zero object.

  4. A category with a product and a coproduct.

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Correct Option: A
Explanation:

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?

  1. A category with a tensor product and an internal hom functor.

  2. A category with a unit object and a zero object.

  3. A category with a product and a coproduct.

  4. A category with a power object.

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Correct Option: A
Explanation:

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?

  1. A category that is locally cartesian closed.

  2. A category that is cartesian closed.

  3. A category that is monoidal closed.

  4. A category that is symmetric monoidal closed.

Show answer
Correct Option: A
Explanation:

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?

  1. A result that relates functors to natural transformations.

  2. A result that relates categories to sets.

  3. A result that relates functors to categories.

  4. A result that relates natural transformations to categories.

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Correct Option: A
Explanation:

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?

  1. A result that shows that the category of categories is cartesian closed.

  2. A result that shows that the category of categories is monoidal closed.

  3. A result that shows that the category of categories is symmetric monoidal closed.

  4. A result that shows that the category of categories is locally cartesian closed.

Show answer
Correct Option: A
Explanation:

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?

  1. A result that shows that every topos is equivalent to a category of sheaves.

  2. A result that shows that every category of sheaves is equivalent to a topos.

  3. A result that shows that every topos is equivalent to a category of presheaves.

  4. A result that shows that every category of presheaves is equivalent to a topos.

Show answer
Correct Option: A
Explanation:

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?

  1. A result that relates Boolean algebras to topological spaces.

  2. A result that relates Boolean algebras to sets.

  3. A result that relates Boolean algebras to categories.

  4. A result that relates Boolean algebras to natural transformations.

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Correct Option: A
Explanation:

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?

  1. A result that embeds the category of small categories into the category of sets.

  2. A result that embeds the category of small categories into the category of categories.

  3. A result that embeds the category of small categories into the category of functors.

  4. A result that embeds the category of small categories into the category of natural transformations.

Show answer
Correct Option: A
Explanation:

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?

  1. A result that shows that the category of categories is symmetric monoidal closed.

  2. A result that shows that the category of categories is cartesian closed.

  3. A result that shows that the category of categories is monoidal closed.

  4. A result that shows that the category of categories is locally cartesian closed.

Show answer
Correct Option: A
Explanation:

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?

  1. A set of axioms that characterize homology theory.

  2. A set of axioms that characterize cohomology theory.

  3. A set of axioms that characterize homotopy theory.

  4. A set of axioms that characterize category theory.

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Correct Option: A
Explanation:

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?

  1. A functor that extends a functor from a small category to a larger category.

  2. A functor that extends a functor from a large category to a smaller category.

  3. A functor that extends a functor from a category to a set.

  4. A functor that extends a functor from a set to a category.

Show answer
Correct Option: A
Explanation:

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.

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