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Category Theory and Computer Science

Description: This quiz covers the fundamental concepts and applications of Category Theory in Computer Science.
Number of Questions: 15
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Tags: category theory computer science mathematics
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In Category Theory, what is a functor?

  1. A mapping between categories that preserves structure

  2. A function between sets that preserves order

  3. A relation between elements of a category

  4. A transformation between functors

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Correct Option: A
Explanation:

A functor is a structure-preserving mapping between categories, consisting of mappings between objects and morphisms that respect composition and identities.

What is the Yoneda Lemma in Category Theory?

  1. Every functor from a category to the category of sets is uniquely determined by its values on the objects of the category

  2. Every category is equivalent to a category of sets

  3. Every functor is an isomorphism

  4. Every category has a unique initial object

Show answer
Correct Option: A
Explanation:

The Yoneda Lemma establishes a fundamental connection between categories and functors, providing a way to represent functors as natural transformations.

What is a monad in Category Theory?

  1. A functor from a category to itself that preserves composition

  2. A functor from a category to the category of sets that preserves structure

  3. A natural transformation between two functors

  4. A category with a unique initial and terminal object

Show answer
Correct Option: A
Explanation:

A monad is an endofunctor that captures the notion of a computational effect, allowing for the sequencing and composition of effects in a structured manner.

What is the Curry-Howard correspondence in Category Theory?

  1. A relationship between types and propositions in logic

  2. A connection between categories and sets

  3. A duality between functors and natural transformations

  4. An equivalence between monads and algebraic structures

Show answer
Correct Option: A
Explanation:

The Curry-Howard correspondence establishes a deep connection between the worlds of logic and computation, showing that types in programming languages correspond to propositions in logic.

What is a category in Category Theory?

  1. A collection of objects and morphisms between them

  2. A set of elements and relations between them

  3. A function between two sets

  4. A transformation between two functors

Show answer
Correct Option: A
Explanation:

A category consists of a collection of objects and morphisms (arrows) between them, satisfying certain axioms related to composition and identity morphisms.

What is a natural transformation in Category Theory?

  1. A morphism between two functors

  2. A function between two sets that preserves structure

  3. A relation between elements of a category

  4. A transformation between categories

Show answer
Correct Option: A
Explanation:

A natural transformation is a morphism between two functors that respects the structure of the categories involved, preserving composition and identities.

What is an initial object in a category?

  1. An object with a unique morphism from every other object

  2. An object with a unique morphism to every other object

  3. An object with no incoming morphisms

  4. An object with no outgoing morphisms

Show answer
Correct Option: A
Explanation:

An initial object in a category is an object that has a unique morphism from every other object in the category, representing a universal starting point.

What is a terminal object in a category?

  1. An object with a unique morphism from every other object

  2. An object with a unique morphism to every other object

  3. An object with no incoming morphisms

  4. An object with no outgoing morphisms

Show answer
Correct Option: B
Explanation:

A terminal object in a category is an object that has a unique morphism to every other object in the category, representing a universal ending point.

What is a product in a category?

  1. An object that represents the Cartesian product of two objects

  2. An object that represents the disjoint union of two objects

  3. An object that represents the intersection of two objects

  4. An object that represents the symmetric difference of two objects

Show answer
Correct Option: A
Explanation:

A product in a category is an object that represents the Cartesian product of two objects, capturing the notion of pairing and combination.

What is a coproduct in a category?

  1. An object that represents the Cartesian product of two objects

  2. An object that represents the disjoint union of two objects

  3. An object that represents the intersection of two objects

  4. An object that represents the symmetric difference of two objects

Show answer
Correct Option: B
Explanation:

A coproduct in a category is an object that represents the disjoint union of two objects, capturing the notion of combining objects without shared elements.

What is an equalizer in a category?

  1. An object that represents the intersection of two morphisms

  2. An object that represents the union of two morphisms

  3. An object that represents the composition of two morphisms

  4. An object that represents the difference of two morphisms

Show answer
Correct Option: A
Explanation:

An equalizer in a category is an object that represents the intersection of two morphisms, capturing the notion of finding common elements between two sets.

What is a coequalizer in a category?

  1. An object that represents the intersection of two morphisms

  2. An object that represents the union of two morphisms

  3. An object that represents the composition of two morphisms

  4. An object that represents the difference of two morphisms

Show answer
Correct Option: B
Explanation:

A coequalizer in a category is an object that represents the union of two morphisms, capturing the notion of combining elements from two sets.

What is a limit in a category?

  1. An object that represents the intersection of a diagram of objects

  2. An object that represents the union of a diagram of objects

  3. An object that represents the composition of a diagram of objects

  4. An object that represents the difference of a diagram of objects

Show answer
Correct Option: A
Explanation:

A limit in a category is an object that represents the intersection of a diagram of objects, capturing the notion of finding common elements among multiple sets.

What is a colimit in a category?

  1. An object that represents the intersection of a diagram of objects

  2. An object that represents the union of a diagram of objects

  3. An object that represents the composition of a diagram of objects

  4. An object that represents the difference of a diagram of objects

Show answer
Correct Option: B
Explanation:

A colimit in a category is an object that represents the union of a diagram of objects, capturing the notion of combining elements from multiple sets.

What is an adjunction in Category Theory?

  1. A pair of functors between two categories that are related by natural isomorphisms

  2. A pair of functors between two categories that are related by natural transformations

  3. A pair of functors between two categories that are related by equivalences

  4. A pair of functors between two categories that are related by isomorphisms

Show answer
Correct Option: A
Explanation:

An adjunction in Category Theory consists of a pair of functors between two categories that are related by natural isomorphisms, capturing the notion of a Galois connection.

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