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Category Theory Fundamentals

Tests knowledge of fundamental category theory concepts including functors, natural transformations, categories, and various mathematical categories

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following is a fundamental concept in category theory?

  1. Functors
  2. Natural Transformations
  3. Categories
  4. All of the above
Question 2 Multiple Choice (Single Answer)

What is a functor?

  1. A mapping between two categories
  2. A mapping between two sets
  3. A mapping between two groups
  4. A mapping between two rings
Question 3 Multiple Choice (Single Answer)

What is a natural transformation?

  1. A morphism between two functors
  2. A morphism between two categories
  3. A morphism between two sets
  4. A morphism between two groups
Question 4 Multiple Choice (Single Answer)

What is a category?

  1. A collection of objects and morphisms
  2. A collection of sets and functions
  3. A collection of groups and homomorphisms
  4. A collection of rings and homomorphisms
Question 5 Multiple Choice (Single Answer)

What is the Yoneda lemma?

  1. A result that relates functors to natural transformations
  2. A result that relates categories to functors
  3. A result that relates objects to morphisms
  4. A result that relates morphisms to objects
Question 6 Multiple Choice (Single Answer)

What is the adjoint functor theorem?

  1. A result that gives conditions for the existence of an adjoint functor
  2. A result that gives conditions for the uniqueness of an adjoint functor
  3. A result that gives conditions for the composition of adjoint functors
  4. A result that gives conditions for the inverse of an adjoint functor
Question 7 Multiple Choice (Single Answer)

What is the category of small categories?

  1. The category whose objects are small categories and whose morphisms are functors
  2. The category whose objects are small categories and whose morphisms are natural transformations
  3. The category whose objects are small categories and whose morphisms are isomorphisms
  4. The category whose objects are small categories and whose morphisms are equivalences
Question 8 Multiple Choice (Single Answer)

What is the category of sets?

  1. The category whose objects are sets and whose morphisms are functions
  2. The category whose objects are sets and whose morphisms are natural transformations
  3. The category whose objects are sets and whose morphisms are isomorphisms
  4. The category whose objects are sets and whose morphisms are equivalences
Question 9 Multiple Choice (Single Answer)

What is the category of groups?

  1. The category whose objects are groups and whose morphisms are homomorphisms
  2. The category whose objects are groups and whose morphisms are natural transformations
  3. The category whose objects are groups and whose morphisms are isomorphisms
  4. The category whose objects are groups and whose morphisms are equivalences
Question 10 Multiple Choice (Single Answer)

What is the category of rings?

  1. The category whose objects are rings and whose morphisms are homomorphisms
  2. The category whose objects are rings and whose morphisms are natural transformations
  3. The category whose objects are rings and whose morphisms are isomorphisms
  4. The category whose objects are rings and whose morphisms are equivalences
Question 11 Multiple Choice (Single Answer)

What is the category of topological spaces?

  1. The category whose objects are topological spaces and whose morphisms are continuous maps
  2. The category whose objects are topological spaces and whose morphisms are natural transformations
  3. The category whose objects are topological spaces and whose morphisms are homeomorphisms
  4. The category whose objects are topological spaces and whose morphisms are equivalences
Question 12 Multiple Choice (Single Answer)

What is the category of smooth manifolds?

  1. The category whose objects are smooth manifolds and whose morphisms are smooth maps
  2. The category whose objects are smooth manifolds and whose morphisms are natural transformations
  3. The category whose objects are smooth manifolds and whose morphisms are diffeomorphisms
  4. The category whose objects are smooth manifolds and whose morphisms are equivalences
Question 13 Multiple Choice (Single Answer)

What is the category of schemes?

  1. The category whose objects are schemes and whose morphisms are morphisms of schemes
  2. The category whose objects are schemes and whose morphisms are natural transformations
  3. The category whose objects are schemes and whose morphisms are isomorphisms
  4. The category whose objects are schemes and whose morphisms are equivalences
Question 14 Multiple Choice (Single Answer)

What is the category of algebraic varieties?

  1. The category whose objects are algebraic varieties and whose morphisms are morphisms of algebraic varieties
  2. The category whose objects are algebraic varieties and whose morphisms are natural transformations
  3. The category whose objects are algebraic varieties and whose morphisms are isomorphisms
  4. The category whose objects are algebraic varieties and whose morphisms are equivalences
Question 15 Multiple Choice (Single Answer)

What is the category of Lie algebras?

  1. The category whose objects are Lie algebras and whose morphisms are homomorphisms of Lie algebras
  2. The category whose objects are Lie algebras and whose morphisms are natural transformations
  3. The category whose objects are Lie algebras and whose morphisms are isomorphisms
  4. The category whose objects are Lie algebras and whose morphisms are equivalences
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