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Category Theory and Geometry

Category Theory and Geometry Quiz

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

In category theory, what is a functor?

  1. A mapping between categories that preserves structure
  2. A function between sets that preserves structure
  3. A relation between elements of a category
  4. A transformation between functors
Question 2 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 sets
  3. A result that relates functors to groups
  4. A result that relates categories to topological spaces
Question 3 Multiple Choice (Single Answer)

What is a category of sheaves?

  1. A category whose objects are sheaves on a topological space
  2. A category whose objects are presheaves on a topological space
  3. A category whose objects are functors from a topological space to the category of sets
  4. A category whose objects are natural transformations between functors
Question 4 Multiple Choice (Single Answer)

What is a topos?

  1. A category that is equivalent to the category of sheaves on a topological space
  2. A category that is equivalent to the category of presheaves on a topological space
  3. A category that is equivalent to the category of functors from a topological space to the category of sets
  4. A category that is equivalent to the category of natural transformations between functors
Question 5 Multiple Choice (Single Answer)

What is a geometric realization of a category?

  1. A functor from a category to the category of topological spaces
  2. A functor from a category to the category of sets
  3. A functor from a category to the category of groups
  4. A functor from a category to the category of natural transformations
Question 6 Multiple Choice (Single Answer)

What is a simplicial complex?

  1. A collection of simplices glued together along their faces
  2. A collection of vertices, edges, and faces
  3. A collection of points, lines, and planes
  4. A collection of sets and functions
Question 7 Multiple Choice (Single Answer)

What is a homology group?

  1. A group that is associated to a simplicial complex
  2. A group that is associated to a topological space
  3. A group that is associated to a group
  4. A group that is associated to a natural transformation
Question 8 Multiple Choice (Single Answer)

What is a cohomology group?

  1. A group that is associated to a simplicial complex
  2. A group that is associated to a topological space
  3. A group that is associated to a group
  4. A group that is associated to a natural transformation
Question 9 Multiple Choice (Single Answer)

What is a fiber bundle?

  1. A surjective map between topological spaces with a special structure
  2. A bijective map between topological spaces with a special structure
  3. An injective map between topological spaces with a special structure
  4. A continuous map between topological spaces with a special structure
Question 10 Multiple Choice (Single Answer)

What is a vector bundle?

  1. A fiber bundle whose fibers are vector spaces
  2. A fiber bundle whose fibers are groups
  3. A fiber bundle whose fibers are sets
  4. A fiber bundle whose fibers are natural transformations
Question 11 Multiple Choice (Single Answer)

What is a tangent bundle?

  1. A vector bundle whose fibers are tangent spaces
  2. A vector bundle whose fibers are normal spaces
  3. A vector bundle whose fibers are cotangent spaces
  4. A vector bundle whose fibers are natural transformations
Question 12 Multiple Choice (Single Answer)

What is a cotangent bundle?

  1. A vector bundle whose fibers are cotangent spaces
  2. A vector bundle whose fibers are tangent spaces
  3. A vector bundle whose fibers are normal spaces
  4. A vector bundle whose fibers are natural transformations
Question 13 Multiple Choice (Single Answer)

What is a symplectic manifold?

  1. A manifold with a symplectic form
  2. A manifold with a Riemannian form
  3. A manifold with a Lorentzian form
  4. A manifold with a Kähler form
Question 14 Multiple Choice (Single Answer)

What is a Kähler manifold?

  1. A manifold with a Kähler form
  2. A manifold with a symplectic form
  3. A manifold with a Riemannian form
  4. A manifold with a Lorentzian form
Question 15 Multiple Choice (Single Answer)

What is a Riemannian manifold?

  1. A manifold with a Riemannian form
  2. A manifold with a symplectic form
  3. A manifold with a Lorentzian form
  4. A manifold with a Kähler form
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