Homotopy Theory and Category Theory
| Description: This quiz covers fundamental concepts and theorems in Homotopy Theory and Category Theory, exploring the relationship between topological spaces and algebraic structures. | |
| Number of Questions: 14 | |
| Created by: Aliensbrain Bot | |
| Tags: homotopy theory category theory topology algebra |
What is the fundamental group of a topological space?
-
The group of all continuous maps from the space to the circle.
-
The group of all homotopy classes of loops in the space.
-
The group of all homology classes in the space.
-
The group of all singular homology classes in the space.
The fundamental group of a topological space is the group of all homotopy classes of loops in the space, based at a fixed point.
What is a category?
-
A collection of objects and arrows between them.
-
A collection of sets and functions between them.
-
A collection of groups and homomorphisms between them.
-
A collection of rings and ring homomorphisms between them.
A category is a collection of objects and arrows between them, where the arrows satisfy certain composition rules.
What is a functor?
-
A map between two categories that preserves the structure.
-
A map between two sets that preserves the structure.
-
A map between two groups that preserves the structure.
-
A map between two rings that preserves the structure.
A functor is a map between two categories that preserves the structure, i.e., it maps objects to objects and arrows to arrows in a way that respects composition.
What is the Yoneda lemma?
-
A result that relates the category of presheaves on a category to the category itself.
-
A result that relates the category of sheaves on a category to the category itself.
-
A result that relates the category of groups to the category of sets.
-
A result that relates the category of rings to the category of modules.
The Yoneda lemma is a result that relates the category of presheaves on a category to the category itself, providing a way to represent objects in the category as functors.
What is the Eilenberg-Steenrod axiom system for homology?
-
A set of axioms that characterize the homology groups of a topological space.
-
A set of axioms that characterize the cohomology groups of a topological space.
-
A set of axioms that characterize the homotopy groups of a topological space.
-
A set of axioms that characterize the singular homology groups of a topological space.
The Eilenberg-Steenrod axiom system for homology is a set of axioms that characterize the homology groups of a topological space, providing a way to define homology groups for arbitrary spaces.
What is the Dold-Kan correspondence?
-
A correspondence between simplicial sets and chain complexes.
-
A correspondence between topological spaces and chain complexes.
-
A correspondence between categories and chain complexes.
-
A correspondence between functors and chain complexes.
The Dold-Kan correspondence is a correspondence between simplicial sets and chain complexes, providing a way to translate between these two structures.
What is the Whitehead theorem?
-
A theorem that characterizes the homotopy groups of a product of two spaces.
-
A theorem that characterizes the homology groups of a product of two spaces.
-
A theorem that characterizes the cohomology groups of a product of two spaces.
-
A theorem that characterizes the singular homology groups of a product of two spaces.
The Whitehead theorem is a theorem that characterizes the homotopy groups of a product of two spaces, providing a way to compute the homotopy groups of a product space from the homotopy groups of its factors.
What is the Hurewicz theorem?
-
A theorem that relates the homology groups of a space to its homotopy groups.
-
A theorem that relates the cohomology groups of a space to its homotopy groups.
-
A theorem that relates the homology groups of a space to its singular homology groups.
-
A theorem that relates the cohomology groups of a space to its singular cohomology groups.
The Hurewicz theorem is a theorem that relates the homology groups of a space to its homotopy groups, providing a way to obtain information about the homology groups of a space from its homotopy groups.
What is the Serre spectral sequence?
-
A spectral sequence that relates the homology groups of a fibration to the homology groups of its base and fiber.
-
A spectral sequence that relates the cohomology groups of a fibration to the cohomology groups of its base and fiber.
-
A spectral sequence that relates the homology groups of a cofibration to the homology groups of its base and fiber.
-
A spectral sequence that relates the cohomology groups of a cofibration to the cohomology groups of its base and fiber.
The Serre spectral sequence is a spectral sequence that relates the homology groups of a fibration to the homology groups of its base and fiber, providing a way to compute the homology groups of a fibration from the homology groups of its base and fiber.
What is the Adams spectral sequence?
-
A spectral sequence that relates the stable homotopy groups of a space to its homology groups.
-
A spectral sequence that relates the stable cohomology groups of a space to its homology groups.
-
A spectral sequence that relates the stable homotopy groups of a space to its singular homology groups.
-
A spectral sequence that relates the stable cohomology groups of a space to its singular cohomology groups.
The Adams spectral sequence is a spectral sequence that relates the stable homotopy groups of a space to its homology groups, providing a way to compute the stable homotopy groups of a space from its homology groups.
What is the Atiyah-Hirzebruch spectral sequence?
-
A spectral sequence that relates the cohomology groups of a complex manifold to its Dolbeault cohomology groups.
-
A spectral sequence that relates the homology groups of a complex manifold to its Dolbeault cohomology groups.
-
A spectral sequence that relates the cohomology groups of a complex manifold to its singular cohomology groups.
-
A spectral sequence that relates the homology groups of a complex manifold to its singular homology groups.
The Atiyah-Hirzebruch spectral sequence is a spectral sequence that relates the cohomology groups of a complex manifold to its Dolbeault cohomology groups, providing a way to compute the cohomology groups of a complex manifold from its Dolbeault cohomology groups.
What is the Grothendieck-Riemann-Roch theorem?
-
A theorem that relates the Euler characteristic of a complex manifold to its Dolbeault cohomology groups.
-
A theorem that relates the Betti numbers of a complex manifold to its Dolbeault cohomology groups.
-
A theorem that relates the Hodge numbers of a complex manifold to its Dolbeault cohomology groups.
-
A theorem that relates the Chern numbers of a complex manifold to its Dolbeault cohomology groups.
The Grothendieck-Riemann-Roch theorem is a theorem that relates the Euler characteristic of a complex manifold to its Dolbeault cohomology groups, providing a way to compute the Euler characteristic of a complex manifold from its Dolbeault cohomology groups.
What is the Thom isomorphism theorem?
-
A theorem that relates the homology groups of a sphere bundle to the homology groups of its base.
-
A theorem that relates the cohomology groups of a sphere bundle to the cohomology groups of its base.
-
A theorem that relates the homology groups of a sphere bundle to its singular homology groups.
-
A theorem that relates the cohomology groups of a sphere bundle to its singular cohomology groups.
The Thom isomorphism theorem is a theorem that relates the homology groups of a sphere bundle to the homology groups of its base, providing a way to compute the homology groups of a sphere bundle from the homology groups of its base.
What is the Bott periodicity theorem?
-
A theorem that relates the stable homotopy groups of spheres to the stable homotopy groups of complex projective spaces.
-
A theorem that relates the stable cohomology groups of spheres to the stable cohomology groups of complex projective spaces.
-
A theorem that relates the stable homology groups of spheres to the stable singular homology groups of complex projective spaces.
-
A theorem that relates the stable cohomology groups of spheres to the stable singular cohomology groups of complex projective spaces.
The Bott periodicity theorem is a theorem that relates the stable homotopy groups of spheres to the stable homotopy groups of complex projective spaces, providing a way to compute the stable homotopy groups of spheres from the stable homotopy groups of complex projective spaces.