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Initial and terminal objects: Quiz

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Question 1: In the ________ with unity and unity-perserving morphisms, the ring of integers Z is an initial object.
Initial and terminal objectsAdjoint functorsCategory of ringsRing (mathematics)

Question 2: If X is a topological space (viewed as a category as above) and C is some small category, we can form the category of all ________ from X to C, using natural transformations as morphisms.
FunctorAdjoint functorsVector spaceGroup (mathematics)

Question 3: A universal morphism from an object X to a functor U can be defined as an initial object in the ________ (XU).
Limit (category theory)Comma categoryAdjoint functorsCategory (mathematics)

Question 4: In the subcategory of ________, however, every trivial monoid (consisting of only the identity element) is a zero object.
Algebraic structureMonoidRing (mathematics)Group (mathematics)

Question 5: In ________, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism IX.
Algebraic topologySet theoryMathematical logicCategory theory

Question 6: For example, the initial object in any ________ with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits).
Concrete categoryGroup actionFull and faithful functorsCategory (mathematics)

Question 7: The category of ________ does not have a terminal object.
Graph theoryPetersen graphSymmetric graphGraph (mathematics)

Question 8: Dually, an initial object is a ________ of the empty diagram ∅ → C and can be thought of as an empty coproduct or categorical sum.
Universal propertyAdjoint functorsInitial and terminal objectsLimit (category theory)

Question 9: Similarly, the category of all small categories with ________ as morphisms has the empty category as initial object and the category 1 (with a single object and morphism) as terminal object.
Group (mathematics)Adjoint functorsFunctorVector space

Question 10: It follows that any ________ which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects.
FunctorVector spaceGroup (mathematics)Adjoint functors







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