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: