User:Fropuff/Questions
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Algebra
[edit]Category theory
[edit]- When is a monoid, considered as a category with one object, a monoidal category?
- When is it symmetric monoidal?
- When is it closed monoidal?
- Kelly claims that "the strict monoidal category of small endofunctors of a well-behaved large category such as Set" is a closed monoidal category that is not biclosed.
- What is a small endofunctor?
- What is the internal Hom functor in this case? Given endofunctors of Set we need a endofunctor such that .
Topology
[edit]- Is the category of topological spaces a closed category with the compact-open topology on function spaces? For any suitable topology?
- not every semiregular space is preregular. is every semiregular space an R0-space? counterexample?
- minimal Hausdorff spaces need not be compact. do they imply any compactness properties? in particular, are minimal Hausdorff spaces locally compact?
- is every KC space sober?
- what spaces have the property: every compact subset is relatively compact
- what spaces have the property: every relatively compact subset is compact
- are subspaces of preregular spaces preregular?
- explore and understand local topological properties
- for every topological property P, define locally-P as follows: a space is locally-P iff every point has a local base of neighborhoods with property P.
- when does P imply locally-P?
- what properties are local in the sense that P = locally-P
- if P => Q does locally-P => locally-Q?
- more generally, when does the existence of a neighborhood with property P imply the existence of a neighborhood base with property P?
- in particular, is this true if property P is inherited by all open subspaces.
- define a locally preregular space. show that every preregular space is locally preregular, and show that the different variants of a local compactness all agree for locally preregular spaces.
- is semi-regularity local? is every locally euclidean space semi-regular?
- understand Kolmogorov quotients with respect to topological properties
- given property P which implies T0, define Q as follows: a space X has property Q iff the Kolmogorov quotient of X has property P
- given property Q which does not imply T0 define P as Q and T0.
- is P = P’
- is Q = Q’
- if P1 => P2 does Q1 => Q2
- if Q1 => Q2 does P1 => P2
- how does locality interact with Kolmogorov quotients
- understand the contravariant adjunction between real algebras and topological spaces (mapping spaces to their algebra of real-valued continuous functions , and mapping algebras to their real dual spaces with the topology of pointwise convergence)
- the algebraic unit is injective iff the algebra is geometric. under what conditions is it surjective?
- show that the set of fixed homomorphisms (those whose kernel has a nonempty zero set) is dense
- show that the topology on agrees with that induced by the zariski topology
- the Zariski topology on the spectrum of a ring
- why is it compact?
- is maxSpec(R) compact? (yes)
- is maxSpec(R) closed or open in Spec(R)? (typically neither)
- show that a subset X of Spec(R) is compact if every free ideal wrt X is finitely-generated
- characterize the irreducible closed sets in Spec(R)
- when is maxSpec Hausdorff? Tychonoff?
Topological groups
[edit]- is a separable, Lindelöf topological group necessarily second-countable?
- is the category of uniform groups (i.e. a group object in the category of uniform spaces) equivalent to the category of balanced groups?
- is the Kolmogorov quotient a left adjoint to the inclusion functor ?
- does left uniform separation of sets imply right uniform separation, and vice versa?
- if is a closed subgroup of , is a (locally trivial) principal H-bundle?
- when is this bundle trivial?
- same questions in the smooth category for Lie groups
- See: [1]