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Natural Transformations

A quiz on natural transformations and related concepts in category theory, including functors, isomorphisms, and extensions

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is a natural transformation?

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

Given two categories $\mathcal{C}$ and $\mathcal{D}$, and two functors $F, G: \mathcal{C} \rightarrow \mathcal{D}$, what is a natural transformation $\eta: F \rightarrow G$?

  1. A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = G(x)$ for all $x \in \mathcal{C}$
  2. A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = G(\eta(x))$ for all $x \in \mathcal{C}$
  3. A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = \eta(G(x))$ for all $x \in \mathcal{C}$
  4. A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = G(F(x))$ for all $x \in \mathcal{C}$
Question 3 Multiple Choice (Single Answer)

What is the condition for a natural transformation $\eta: F \rightarrow G$ to be an isomorphism?

  1. $F$ and $G$ are isomorphic functors
  2. $F$ and $G$ are equivalent functors
  3. $F$ and $G$ are naturally equivalent functors
  4. $F$ and $G$ are adjoint functors
Question 4 Multiple Choice (Single Answer)

Given two categories $\mathcal{C}$ and $\mathcal{D}$, and two functors $F, G: \mathcal{C} \rightarrow \mathcal{D}$, what is a natural isomorphism $\eta: F \rightarrow G$?

  1. A natural transformation $\eta: F \rightarrow G$ that is an isomorphism
  2. A natural transformation $\eta: F \rightarrow G$ that is a bijection
  3. A natural transformation $\eta: F \rightarrow G$ that is a homeomorphism
  4. A natural transformation $\eta: F \rightarrow G$ that is a diffeomorphism
Question 5 Multiple Choice (Single Answer)

What is the Yoneda lemma?

  1. A result in category theory that relates functors to natural transformations
  2. A result in category theory that relates categories to functors
  3. A result in category theory that relates functors to adjunctions
  4. A result in category theory that relates categories to natural transformations
Question 6 Multiple Choice (Single Answer)

What is the Beck-Chevalley condition?

  1. A condition on a natural transformation that ensures that it is an isomorphism
  2. A condition on a natural transformation that ensures that it is a bijection
  3. A condition on a natural transformation that ensures that it is a homeomorphism
  4. A condition on a natural transformation that ensures that it is a diffeomorphism
Question 7 Multiple Choice (Single Answer)

What is the coherence theorem for natural transformations?

  1. A theorem that states that any two natural transformations between the same functors are equal
  2. A theorem that states that any two natural transformations between the same functors are isomorphic
  3. A theorem that states that any two natural transformations between the same functors are equivalent
  4. A theorem that states that any two natural transformations between the same functors are naturally equivalent
Question 8 Multiple Choice (Single Answer)

What is a dinatural transformation?

  1. A natural transformation between two natural transformations
  2. A natural transformation between two functors
  3. A natural transformation between two categories
  4. A natural transformation between two adjunctions
Question 9 Multiple Choice (Single Answer)

What is a profunctor?

  1. A functor between two categories that is contravariant in one argument and covariant in the other
  2. A functor between two categories that is covariant in both arguments
  3. A functor between two categories that is contravariant in both arguments
  4. A functor between two categories that is neither covariant nor contravariant in either argument
Question 10 Multiple Choice (Single Answer)

What is a lax natural transformation?

  1. A natural transformation that satisfies the Beck-Chevalley condition
  2. A natural transformation that satisfies the Yoneda lemma
  3. A natural transformation that satisfies the coherence theorem
  4. A natural transformation that does not satisfy any of the above conditions
Question 11 Multiple Choice (Single Answer)

What is a strict natural transformation?

  1. A natural transformation that satisfies the Beck-Chevalley condition
  2. A natural transformation that satisfies the Yoneda lemma
  3. A natural transformation that satisfies the coherence theorem
  4. A natural transformation that satisfies all of the above conditions
Question 12 Multiple Choice (Single Answer)

What is a pseudonatural transformation?

  1. A natural transformation that is not strict
  2. A natural transformation that is lax
  3. A natural transformation that is dinatural
  4. A natural transformation that is profunctorial
Question 13 Multiple Choice (Single Answer)

What is a natural isomorphism?

  1. A natural transformation that is an isomorphism
  2. A natural transformation that is a bijection
  3. A natural transformation that is a homeomorphism
  4. A natural transformation that is a diffeomorphism
Question 14 Multiple Choice (Single Answer)

What is a Yoneda extension?

  1. A functor that extends a functor from a category to its category of presheaves
  2. A functor that extends a functor from a category to its category of sheaves
  3. A functor that extends a functor from a category to its category of toposes
  4. A functor that extends a functor from a category to its category of groupoids
Question 15 Multiple Choice (Single Answer)

What is a Kan extension?

  1. A functor that extends a functor from a category to its category of Kan complexes
  2. A functor that extends a functor from a category to its category of simplicial complexes
  3. A functor that extends a functor from a category to its category of CW complexes
  4. A functor that extends a functor from a category to its category of topological spaces
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