Adjoints and Monads

Question 1 Multiple Choice (Single Answer)

Given categories (C) and (D), what is the definition of an adjoint pair of functors (F: C \to D) and (G: D \to C)?

For every object \(X \in C\) and every object \(Y \in D\), there exists a bijection between the set of morphisms from \(F(X)\) to \(Y\) in \(D\) and the set of morphisms from \(X\) to \(G(Y)\) in \(C\).
For every object \(X \in C\) and every object \(Y \in D\), there exists a natural isomorphism between the functor \(F\) and the functor \(G\).
For every object \(X \in C\) and every object \(Y \in D\), there exists a natural transformation from \(F\) to \(G\) and a natural transformation from \(G\) to \(F\) such that their composition is the identity natural transformation.
For every object \(X \in C\) and every object \(Y \in D\), there exists a natural transformation from \(F\) to \(G\) such that its composition with itself is the identity natural transformation.

Question 2 Multiple Choice (Single Answer)

In the context of an adjoint pair of functors (F: C \to D) and (G: D \to C), what is the unit of the adjunction?

A natural transformation \(\eta: \text{Id}_C \to G \circ F\) such that for every object \(X \in C\), the component \(\eta_X: X \to G(F(X))\) is an isomorphism.
A natural transformation \(\eta: F \circ G \to \text{Id}_D\) such that for every object \(Y \in D\), the component \(\eta_Y: F(G(Y)) \to Y\) is an isomorphism.
A natural transformation \(\eta: \text{Id}_C \to F \circ G\) such that for every object \(X \in C\), the component \(\eta_X: X \to F(G(X))\) is an isomorphism.
A natural transformation \(\eta: G \circ F \to \text{Id}_D\) such that for every object \(Y \in D\), the component \(\eta_Y: G(F(Y)) \to Y\) is an isomorphism.

Question 3 Multiple Choice (Single Answer)

In the context of an adjoint pair of functors (F: C \to D) and (G: D \to C), what is the counit of the adjunction?

A natural transformation \(\epsilon: F \circ G \to \text{Id}_D\) such that for every object \(Y \in D\), the component \(\epsilon_Y: F(G(Y)) \to Y\) is an isomorphism.
A natural transformation \(\epsilon: \text{Id}_C \to G \circ F\) such that for every object \(X \in C\), the component \(\epsilon_X: X \to G(F(X))\) is an isomorphism.
A natural transformation \(\epsilon: \text{Id}_D \to F \circ G\) such that for every object \(Y \in D\), the component \(\epsilon_Y: Y \to F(G(Y))\) is an isomorphism.
A natural transformation \(\epsilon: G \circ F \to \text{Id}_C\) such that for every object \(X \in C\), the component \(\epsilon_X: G(F(X)) \to X\) is an isomorphism.

Question 4 Multiple Choice (Single Answer)

What is the relationship between the unit and counit of an adjoint pair of functors?

The unit and counit are natural transformations that are inverses of each other.
The unit and counit are natural transformations that are composable.
The unit and counit are natural transformations that are equal to each other.
The unit and counit are natural transformations that are orthogonal to each other.

Question 5 Multiple Choice (Single Answer)

Given a category (C), what is a monad on (C)?

A triple \((T, \eta, \mu)\) consisting of a functor \(T: C \to C\), a natural transformation \(\eta: \text{Id}_C \to T\), and a natural transformation \(\mu: T \circ T \to T\) such that \(\mu \circ \mu = \mu \circ T \circ \eta\) and \(\mu \circ \eta \circ T = \eta \circ \mu\).
A triple \((T, \eta, \mu)\) consisting of a functor \(T: C \to C\), a natural transformation \(\eta: T \to \text{Id}_C\), and a natural transformation \(\mu: T \circ T \to T\) such that \(\mu \circ \mu = \mu \circ T \circ \eta\) and \(\mu \circ \eta \circ T = \eta \circ \mu\).
A triple \((T, \eta, \mu)\) consisting of a functor \(T: C \to C\), a natural transformation \(\eta: \text{Id}_C \to T\), and a natural transformation \(\mu: T \circ T \to T\) such that \(\mu \circ \mu = \mu \circ \eta \circ T\) and \(\mu \circ \eta \circ T = \eta \circ \mu\).
A triple \((T, \eta, \mu)\) consisting of a functor \(T: C \to C\), a natural transformation \(\eta: T \to \text{Id}_C\), and a natural transformation \(\mu: T \circ T \to T\) such that \(\mu \circ \mu = \mu \circ T \circ \eta\) and \(\mu \circ \eta \circ T = \eta \circ \mu\).

Question 6 Multiple Choice (Single Answer)

What is the relationship between monads and adjoint pairs of functors?

Every monad can be represented as an adjoint pair of functors.
Every adjoint pair of functors can be represented as a monad.
Monads and adjoint pairs of functors are unrelated concepts.
Monads and adjoint pairs of functors are equivalent concepts.

Question 7 Multiple Choice (Single Answer)

What is the Kleisli category associated with a monad ((T, \eta, \mu))?

The category whose objects are objects of \(C\) and whose morphisms are morphisms of \(C\) that are compatible with the monad structure.
The category whose objects are objects of \(C\) and whose morphisms are natural transformations between functors of the form \(T^n \to T^m\), where \(n\) and \(m\) are natural numbers.
The category whose objects are objects of \(C\) and whose morphisms are natural transformations between functors of the form \(T^n \to T^{n+1}\), where \(n\) is a natural number.
The category whose objects are objects of \(C\) and whose morphisms are natural transformations between functors of the form \(T^n \to T^{n-1}\), where \(n\) is a natural number.

Question 8 Multiple Choice (Single Answer)

What is the Eilenberg-Moore category associated with a monad ((T, \eta, \mu))?

The category whose objects are objects of \(C\) and whose morphisms are natural transformations between functors of the form \(T^n \to T^m\), where \(n\) and \(m\) are natural numbers.
The category whose objects are objects of \(C\) and whose morphisms are natural transformations between functors of the form \(T^n \to T^{n+1}\), where \(n\) is a natural number.
The category whose objects are objects of \(C\) and whose morphisms are natural transformations between functors of the form \(T^n \to T^{n-1}\), where \(n\) is a natural number.
The category whose objects are objects of \(C\) and whose morphisms are morphisms of \(C\) that are compatible with the monad structure.

Question 9 Multiple Choice (Single Answer)

What is the relationship between the Kleisli category and the Eilenberg-Moore category associated with a monad?

The Kleisli category is a subcategory of the Eilenberg-Moore category.
The Eilenberg-Moore category is a subcategory of the Kleisli category.
The Kleisli category and the Eilenberg-Moore category are equivalent categories.
The Kleisli category and the Eilenberg-Moore category are unrelated categories.

Question 10 Multiple Choice (Single Answer)

What is a free monad on a functor (F: C \to C)?

A monad \((T, \eta, \mu)\) such that \(T = F\) and \(\eta\) and \(\mu\) are the identity natural transformations.
A monad \((T, \eta, \mu)\) such that \(T = F\) and \(\eta\) and \(\mu\) are natural transformations that satisfy the monad laws.
A monad \((T, \eta, \mu)\) such that \(T = F \circ F\) and \(\eta\) and \(\mu\) are natural transformations that satisfy the monad laws.
A monad \((T, \eta, \mu)\) such that \(T = F \circ F\) and \(\eta\) and \(\mu\) are the identity natural transformations.

Question 11 Multiple Choice (Single Answer)

What is the relationship between free monads and adjoint pairs of functors?

Every free monad can be represented as an adjoint pair of functors.
Every adjoint pair of functors can be represented as a free monad.
Free monads and adjoint pairs of functors are unrelated concepts.
Free monads and adjoint pairs of functors are equivalent concepts.

Question 12 Multiple Choice (Single Answer)

What is a monadic functor?

A functor \(F: C \to D\) such that there exists a monad \((T, \eta, \mu)\) on \(C\) and a natural transformation \(\phi: F \to T\) that satisfies certain properties.
A functor \(F: C \to D\) such that there exists a monad \((T, \eta, \mu)\) on \(D\) and a natural transformation \(\phi: F \to T\) that satisfies certain properties.
A functor \(F: C \to D\) such that there exists a monad \((T, \eta, \mu)\) on \(C\) and a natural transformation \(\phi: T \to F\) that satisfies certain properties.
A functor \(F: C \to D\) such that there exists a monad \((T, \eta, \mu)\) on \(D\) and a natural transformation \(\phi: T \to F\) that satisfies certain properties.

Question 13 Multiple Choice (Single Answer)

What is the relationship between monadic functors and adjoint pairs of functors?

Every monadic functor can be represented as an adjoint pair of functors.
Every adjoint pair of functors can be represented as a monadic functor.
Monadic functors and adjoint pairs of functors are unrelated concepts.
Monadic functors and adjoint pairs of functors are equivalent concepts.

Question 14 Multiple Choice (Single Answer)

What is a Kleisli triple?

A triple \((K, \eta, \mu)\) consisting of a functor \(K: C \to C\), a natural transformation \(\eta: \text{Id}_C \to K\), and a natural transformation \(\mu: K \circ K \to K\) such that \(\mu \circ \mu = \mu \circ K \circ \eta\) and \(\mu \circ \eta \circ K = \eta \circ \mu\).
A triple \((K, \eta, \mu)\) consisting of a functor \(K: C \to C\), a natural transformation \(\eta: K \to \text{Id}_C\), and a natural transformation \(\mu: K \circ K \to K\) such that \(\mu \circ \mu = \mu \circ K \circ \eta\) and \(\mu \circ \eta \circ K = \eta \circ \mu\).
A triple \((K, \eta, \mu)\) consisting of a functor \(K: C \to C\), a natural transformation \(\eta: \text{Id}_C \to K\), and a natural transformation \(\mu: K \circ K \to K\) such that \(\mu \circ \mu = \mu \circ K \circ \eta\) and \(\mu \circ \eta \circ K = \eta \circ \mu\).
A triple \((K, \eta, \mu)\) consisting of a functor \(K: C \to C\), a natural transformation \(\eta: K \to \text{Id}_C\), and a natural transformation \(\mu: K \circ K \to K\) such that \(\mu \circ \mu = \mu \circ K \circ \eta\) and \(\mu \circ \eta \circ K = \eta \circ \mu\).

Question 15 Multiple Choice (Single Answer)

What is the relationship between Kleisli triples and monads?

Every Kleisli triple can be represented as a monad.
Every monad can be represented as a Kleisli triple.
Kleisli triples and monads are unrelated concepts.
Kleisli triples and monads are equivalent concepts.

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