A mapping between two categories that preserves their structure.

A function between two sets that preserves their algebraic structure.

A relation between two objects in a category.

A transformation between two functors.

A result in category theory that establishes a bijection between the category of presheaves on a category and the category of functors from that category to the category of sets.

A theorem in analysis that provides a sufficient condition for a function to be continuous.

A proposition in abstract algebra that characterizes simple groups.

A lemma in topology that relates the homology and cohomology groups of a space.

A category whose objects are sheaves on a topological space.

A category whose objects are presheaves on a topological space.

A category whose objects are functors from a topological space to the category of sets.

A category whose objects are morphisms between sheaves on a topological space.

The cohomology groups associated to the de Rham complex of differential forms on the manifold.

The homology groups associated to the de Rham complex of differential forms on the manifold.

The cohomology groups associated to the singular chain complex of the manifold.

The homology groups associated to the singular chain complex of the manifold.

A theorem in differential geometry that decomposes a differential form on a Riemannian manifold into a sum of exact, coexact, and harmonic forms.

A theorem in analysis that provides a sufficient condition for a function to be differentiable.

A theorem in algebraic topology that characterizes simply connected spaces.

A theorem in number theory that provides a formula for the number of primes less than a given number.

A theorem in differential geometry that relates the index of an elliptic operator on a compact manifold to the topological invariants of the manifold.

A theorem in analysis that provides a sufficient condition for a function to be integrable.

A theorem in algebraic topology that characterizes homology groups of spheres.

A theorem in number theory that provides a formula for the distribution of prime numbers.

A spectral sequence associated to a filtered category.

A spectral sequence associated to a cofiltered category.

A spectral sequence associated to a triangulated category.

A spectral sequence associated to a derived category.

A theorem in algebraic geometry that relates the number of zeros and poles of a meromorphic function on a Riemann surface to the topological invariants of the surface.

A theorem in analysis that provides a sufficient condition for a function to be analytic.

A theorem in number theory that provides a formula for the number of solutions to a Diophantine equation.

A theorem in topology that characterizes compact Hausdorff spaces.

A conjecture in algebraic geometry that relates the de Rham cohomology of a complex projective variety to its singular cohomology.

A conjecture in analysis that provides a sufficient condition for a function to be harmonic.

A conjecture in number theory that provides a formula for the distribution of prime numbers.

A conjecture in topology that characterizes simply connected spaces.

A conjecture in algebraic topology that relates the K-theory of a group to the topology of its classifying space.

A conjecture in analysis that provides a sufficient condition for a function to be differentiable.

A conjecture in number theory that provides a formula for the distribution of prime numbers.

A conjecture in topology that characterizes compact Hausdorff spaces.

A conjecture in algebraic topology that relates the homology of a manifold to the topology of its boundary.

A conjecture in analysis that provides a sufficient condition for a function to be integrable.

A conjecture in number theory that provides a formula for the distribution of prime numbers.

A conjecture in topology that characterizes compact Hausdorff spaces.

A conjecture in algebraic topology that relates the homology of a manifold to the topology of its singular set.

A conjecture in analysis that provides a sufficient condition for a function to be differentiable.

A conjecture in number theory that provides a formula for the distribution of prime numbers.

A conjecture in topology that characterizes compact Hausdorff spaces.

A conjecture in algebraic geometry that relates the cohomology of an algebraic variety to the Galois cohomology of its function field.

A conjecture in analysis that provides a sufficient condition for a function to be analytic.

A conjecture in number theory that provides a formula for the distribution of prime numbers.

A conjecture in topology that characterizes compact Hausdorff spaces.

A vast research program in mathematics that aims to unify various areas of mathematics, including number theory, algebraic geometry, and representation theory.

A conjecture in analysis that provides a sufficient condition for a function to be integrable.

A conjecture in number theory that provides a formula for the distribution of prime numbers.

A conjecture in topology that characterizes compact Hausdorff spaces.