User:Fropuff/Questions

Algebra

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 $X$ $X$ to their algebra of real-valued continuous functions $C(X)$ $C(X)$ , and mapping algebras $A$ $A$ to their real dual spaces $|A|=\operatorname {Hom} (A,R)$ $|A|=\operatorname {Hom} (A,R)$ 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 $H$ $H$ is a closed subgroup of $G$ $G$ , is $G\to G/H$ $G\to G/H$ a (locally trivial) principal H-bundle?
- when is this bundle trivial?
- same questions in the smooth category for Lie groups
- See: [1]

User:Fropuff/Questions

Algebra

Category theory

Topology

Topological groups

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