User:Fropuff/Questions

Category theory

• 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 $F,G,H$ of Set we need a endofunctor $[G,H]$ such that $\operatorname {Nat} (F\circ G,H)\cong \operatorname {Nat} (F,[G,H])$.

Topology

• 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 $X$ to their algebra of real-valued continuous functions $C(X)$, and mapping algebras $A$ to their real dual spaces $|A|=\operatorname {Hom} (A,R)$ 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 $|A|$ 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

• 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 $\mathbf {HausTopGrp} \to \mathbf {TopGrp}$?
• does left uniform separation of sets imply right uniform separation, and vice versa?
• if $H$ is a closed subgroup of $G$, is $G\to G/H$ a (locally trivial) principal H-bundle?
  • when is this bundle trivial?
  • same questions in the smooth category for Lie groups
  • See: [1]