nLab
exponential law for spaces

Contents

Idea

When in a convenient category of topological spaces, e.g. compactly generated spaces, the category is cartesian closed, so that there is an adjunction between the mapping space and the cartesian product in that category. For general topological spaces there is no globally defined adjunction, but we can instead characterize exactly which spaces are exponentiable.

Exponentiable spaces

For CC a category with finite products, recall that an object cc is exponentiable if the functor c×:CCc \times -: C \to C has a right adjoint, usually denoted () c:CC(-)^c: C \to C.

Theorem (Exponentiability, I)

Let Top be the category of all topological spaces. An object XX of TopTop is exponentiable if and only if X×:TopTopX \times -: Top \to Top preserves coequalizers, or equivalently quotient spaces.

This functor always preserves coproducts, so this condition is equivalent to saying that X×X \times - preserves all small colimits. This is then equivalent to exponentiability by the adjoint functor theorem.

This condition, however, is not really any more explicit. More interesting is to characterize the exponentiable spaces in terms of a point-set-topological condition.

Core-compactness

For open subsets UU and VV of a topological space XX, we write VUV\ll U to mean that any open cover of UU admits a finite subcover of VV; this is read as VV is relatively compact under UU or VV is way below UU. We say that XX is core-compact if for every open neighborhood UU of a point xx, there exists an open neighborhood VV of xx with VUV\ll U. In other words, XX is core-compact iff for all open subsets VV, we have V={UUV}V = \bigcup \{ U | U\ll V \}. This says essentially the same thing as saying that the open-set lattice of XX is a continuous lattice, which yields the corresponding definition for locales.

Theorem (Exponentiability, II)

An object XX of TopTop is exponentiable if and only if it is core-compact.

If XX is Hausdorff, then core-compactness is equivalent to local compactness; thus in particular all locally compact Hausdorff spaces are exponentiable. For this reason, that core-compactness is also called quasi local compactness.

When XX is core-compact, we can explicitly describe the exponential topology on Y XY^X (whose points are continuous maps f:XYf: X \to Y). It is generated by subbasis elements O U,VO_{U,V}, for UU an open subset of XX and VV an open subset of YY, where a continuous map f:XYf\colon X \to Y belongs to O U,VO_{U,V} iff Uf 1(V)U\ll f^{-1}(V):

O U,V={fY X:Uf 1(V)}. O_{U, V} = \{f \in Y^X: U \ll f^{-1}(V)\} .

If XX and YY are Hausdorff, then this topology on Y XY^X coincides with the compact-open topology.

In terms of convergence

Exponentiable (i.e. core-compact) spaces can also be characterized in terms of ultrafilter convergence. Recall that a topological space can equivalently be defined as a lax algebra? for the ultrafilter monad UU on the (1,2)-category Rel of sets and relations. In other words, it consists of a set XX and a relation R:UXXR\colon U X \to X called “convergence”, such that id XRηid_X \subseteq R \circ \eta and RURRμR\circ U R \subseteq R\circ \mu, where η\eta and μ\mu are the unit and multiplication of the ultrafilter monad, regarded as relations. In the paper

  • Claudio Pisani?, Convergence in exponentiable spaces, TAC

it is shown that a space is exponentiable (i.e. core-compact) if and only if we have equality in the multiplication law RUR=RμR\circ U R = R\circ \mu.

Some intuition for this characterization can be obtained as follows. Consider the standard non-locally-compact space, the rationals \mathbb{Q} as a subspace of the reals \mathbb{R}. Suppose that xx is a rational number and that y ny_n is a sequence of irrationals converging to xx. Then for each nn we can find a sequence z m nz^n_m of rationals which converges to y ny_n; hence the z m nz^n_m form a “sequence of sequences” which “globally converges” to xx in \mathbb{Q}, i.e. which are related to xx by the composite relation RμR\circ \mu, but for which does not converge elementwise to an intermediate sequence which in turn converges to xx, i.e. it is not related to xx by the relation RURR \circ U R. It turns out that when generalized to ultrafilter convergence, this sort of behavior exactly characterizes what it means to fail to be (quasi) locally compact.

General exponential laws

If XX is exponentiable, then the exponential law gives us an isomorphism of sets Map(Y,B X)Map(X×Y,B)Map(Y,B^X) \cong Map(X\times Y,B) for any other spaces BB and YY. If YY is also exponentiable, then the Yoneda lemma yields from this a homeomorphism B X×Y(B X) YB^{X\times Y} \cong (B^X)^Y. However, we can also say some things in general without all spaces involved being exponentiable.

We now agree to denote by Map(X,Y)=X YMap(X,Y)=X^Y the space of continuous maps XYX\to Y in the compact-open topology.

Theorem (Exponential law)

Let X,Y,BX,Y,B be topological spaces. For any fB X×Yf\in B^{X\times Y}, the formula

[(θf)(y)](x)=f(x,y) [(\theta f)(y)](x) = f(x,y)

defines a continuous map θf:YB X\theta f:Y\to B^X which we call the map adjoint to ff, or the adjunct of ff.

The adjunction map

θ:Map(X×Y,B)Map(Y,B X),θ:fθf \theta : Map(X\times Y,B)\to Map(Y,B^X), \,\,\,\,\,\,\,\theta:f\mapsto \theta f

is a one-to-one function, and if XX is locally compact and Hausdorff then θ\theta is a bijection. Independently from that assumption on XX, if YY is Hausdorff, then θ\theta is continuous in the compact-open topology

θ:B X×Y(B X) Y. \theta : B^{X\times Y}\to (B^X)^Y.

If both assumptions (on XX and YY) are satisfied, then θ\theta is not only a continuous bijection, but also open, hence a homeomorphism.

Based exponential laws

There is also a version for based (= pointed) topological spaces. The cartesian product then needs to be replace by the smash product of the based spaces. Regarding that the maps preserve the base point, the adjunction map θ\theta induces the adjunction map

θ *:Map *(XY,B)Map *(Y,B X)\theta_*:Map_*(X\wedge Y,B)\to Map_*(Y,B^X)

where the mapping space Map *Map_* for based spaces is the subspace of the usual mapping space, in the compact-open topology, which consists of the mappings preserving the base point.

It appears that θ *\theta_* is again one-to-one and continuous, and it is bijective if XX is locally compact Hausdorff. If YY is also Hausdorff then θ *\theta_* is a homeomorphism.

References

Revised on September 13, 2011 20:23:51 by Urs Schreiber (82.93.78.115)