which also shows the aforementioned claim that every monad arises from a free-
forgetful adjunction.

######  Proposition 3.1.3.4 (Universal property of free $T$-modules).

Let $\mathsf{C}$ be a category, let $(T,i,m)$ be a monad on $\mathsf{C}$ and
let $X$ be an object of $\mathsf{C}$. Then for any $\mathsf{C}$-morphism,

$f:X\to A,$

from $X$ to some $T$-module $A$, there is a unique morphism of $T$-modules,

$\overline{f}:TX\to A,$

such that the following diagram commutes:

${X}$${A}$${TX}$$\scriptstyle{f}$$\scriptstyle{i}$$\scriptstyle{\overline{f}}$

In other words, we have an adjunction,

$\operatorname{Hom}_{\mathsf{C}}(X,|A|)\cong\operatorname{Hom}_{\mathsf{C}^{T}}(TX,A),\qquad(X\in\mathsf{C},A\in\mathsf{C}^{T})$

where $|-|:\mathsf{C}^{T}\to\mathsf{C}$ denotes the forgetful functor.

###### Proof.

Take $\overline{f}:=t^{A}\circ Tf$. This makes the desired diagram commute and
thus shows existence. For uniqueness, notice that $\overline{f}$ is
necessarily of this form: being a morphism of $T$-modules,

$\overline{f}\circ m=t^{A}\circ T\overline{f}.$

But then,

$\overline{f}\circ m\circ Ti=t^{A}\circ T\overline{f}\circ Ti.$

Using the unit law, the left hand side is just $\overline{f}$, while the right
hand side equals $t^{A}\circ Tf$, by functoriality of $T$ and commutativity of
the diagram. Thus, $\overline{f}:=t^{A}\circ Tf$, showing uniqueness. ∎

###  3.2. Tensor Products: Monoidal Categories

What does it mean for a category to admit a “good notion of tensor product”?
And how does one construct such tensor products? Questions like these are
naturally treated using the language of _monoidal categories_ (and their
relatives).

####  3.2.1. Monoidal categories

######  Definition 3.2.1.1.

A _monoidal category_ is a category $\mathsf{C}$ together with

  1. 1.

a functor,

$\otimes_{\mathsf{C}}:\mathsf{C}\times\mathsf{C}\to\mathsf{C},$

called the _tensor product_ ,

  2. 2.

an object $I_{\mathsf{C}}$, called the _tensor unit_ ,

  3. 3.

a natural isomorphism,

$\alpha^{A,B,C}:(A\otimes_{\mathsf{C}}B)\otimes_{\mathsf{C}}C\cong
A\otimes_{\mathsf{C}}(B\otimes_{\mathsf{C}}C),\;\;\;\;(A,B,C\in\mathsf{C}$

called the _associator_ ,

  4. 4.

two further natural isomorphisms,

$\lambda^{A}:I_{\mathsf{C}}\otimes_{\mathsf{C}}A\cong
A,\;\;\rho^{A}:A\otimes{\mathsf{C}}I_{\mathsf{C}}\cong
A,\;\;\;(A\in\mathsf{C})$

called the left and right _unitors_ , respectively,

subject to the commutativity of the following two diagrams, the _coherence
conditions_ :

  1. 1.

The triangle identity:

${{(A\otimes_{\mathsf{C}}I_{\mathsf{C}})\otimes
B}}$${{A\otimes_{\mathsf{C}}(I_{\mathsf{C}}\otimes
B)}}$${{A\otimes_{\mathsf{C}}B}}$$\scriptstyle{\alpha^{A,I_{\mathsf{C}},B}}$$\scriptstyle{\rho^{A}\otimes_{\mathsf{C}}I_{B}}$$\scriptstyle{I_{A}\otimes_{\mathsf{C}}\lambda_{B}}$

  2. 2.

The pentagon identity:

${{(A\otimes_{\mathsf{C}}B)\otimes_{\mathsf{C}}(C\otimes_{\mathsf{C}}D)}}$${{(((A\otimes_{\mathsf{C}}B)\otimes_{\mathsf{C}}C)\otimes_{\mathsf{C}}D)}}$${{A\otimes_{\mathsf{C}}(B\otimes_{\mathsf{C}}(C\otimes_{\mathsf{C}}D))}}$${{(A\otimes_{\mathsf{C}}(B\otimes_{\mathsf{C}}C))\otimes_{\mathsf{C}}D}}$${{A\otimes_{\mathsf{C}}((B\otimes_{\mathsf{C}}C)\otimes_{\mathsf{C}}D)}}$$\scriptstyle{\alpha^{A\otimes_{\mathsf{C}}B,C,D}}$$\scriptstyle{\alpha^{A,B,C\otimes_{\mathsf{C}}D}}$$\scriptstyle{\alpha^{A,B\otimes_{\mathsf{C}}C,D}}$$\scriptstyle{\alpha^{A,B,C}\otimes_{\mathsf{C}}D}$$\scriptstyle{A\otimes_{\mathsf{C}}\alpha^{B,C,D}}$

######  Warning 3.2.1.2.

Despite being a crucial part of the structure of a monoidal category, we will
sometimes leave the associator and unitors implicit, as they are often easily
inferred from context. For example, we will use (common) phrases like “let
$(\mathsf{C},\otimes,I)$ be a monoidal category”, specifying the associator
and unitors only when they become relevant. This is (partially) justified by
_Mac Lane’s Coherence Theorem_ (see [ML71, p. 165-170]) which is also the
motivation for the otherwise perhaps somewhat obscure coherence conditions in
the definition of a monoidal category.

######  Example 3.2.1.3.

Let $\mathsf{C}$ be a category with finite products. Then $\mathsf{C}$ is a
monoidal category with respect to the cartesian product $\times$ as the tensor
product, the initial object $1$ as tensor unit, and the associator and unitors
given in the evident way.

######  Example 3.2.1.4.

Let $R$ be a commutative ring and let $R\mathsf{Mod}$ be the category of
modules over $R$. Then $R\mathsf{Mod}$ becomes a monoidal category under the
tensor product $\otimes_{R}$ of $R$-modules, with $R$ serving as the tensor
unit. (As before, we omit an explicit description of the associator and
unitors.) This example illustrates that monoidal _structure_ is not a mere
_property_ of a category: the category of $R\mathsf{Mod}$ is also a monoidal
category with respect to the cartesian product.

##### Monoidal functors and adjunctions

Monoidal categories come with several corresponding notions of _monoidal
functor_ , varying in strength.

######  Definition 3.2.1.5 (lax/strong/strict monoidal functors).

A _lax monoidal functor_ between monoidal categories $\mathsf{C}$ and
$\mathsf{D}$ is a functor $F:\mathsf{C}\to\mathsf{D}$ together with

  1. 1.

a morphism, the _unit_ ,

$I_{\mathsf{D}}\to F(I_{\mathsf{C}}),$

  2. 2.

and a natural transformation, the _multiplication_

$F(A)\otimes F(B)\to F(A\otimes B),\;\;\;\;(A,B\in\mathsf{C})$

subject to a number of coherence conditions (which we omit, see [ML71, pp.
255-56] for a detailed definition). If unit and multiplication are both
isomorphisms, then $F$ is a _strong monoidal functor_ , and if, in addition,
these isomorphisms are even _identities_ , then $F$ is called a _strict
monoidal functor_. Even though a monoidal functor is a functor with additional
_structure_ , we will, as is common, sometimes abuse terminology and treat
this structure as though it were a mere _property_. For example, when we say
that a functor $F$ “is” a monoidal functor, we mean that $F$ admits a specific
structure of a monoidal functor which is left implicit.

Adjunctions also have a monoidal counterpart.

######  Definition 3.2.1.6.

A _monoidal adjunction_ is an adjunction

${{\mathsf{C}}}$${{\mathsf{D}}}$$\scriptstyle{G}$$\scriptstyle{F}$$\scriptstyle{\dashv}$

between monoidal categories $\mathsf{C},\mathsf{D}$ in which the left adjoint
$F$ is a strong monoidal functor.

####  3.2.2. Symmetric monoidal categories

Demanding further structure which renders the tensor product commutative, or
_symmetric_ , in a suitable sense, leads to the following definition.

######  Definition 3.2.2.1.

A symmetric monoidal category is a monoidal category $\mathsf{C}$ together
with a natural isomorphism

$\sigma^{A,B}:A\otimes B\cong B\otimes A,\;\;\;\;(A,B\in\mathsf{C})$

called the _braiding_ , satisfying the identity

$\sigma^{B,A}\circ\sigma^{A,B}=\mathsf{id}_{A\otimes B},$

as well as certain coherence conditions (which we omit, a detailed definition
can be found in [ML71, pp. 252-53]).

##### Symmetric monoidal functors and adjunctions

As for the case of bare monoidal categories, there are associated notions of
_symmetric monoidal functor_ and _symmetric monoidal adjunction_. We will only
need the case of _strong_ symmetric monoidal functors.

######  Definition 3.2.2.2.

A _strong symmetric monoidal functor_ is a strong monoidal functor
$F:\mathsf{C}\to\mathsf{D}$ between symmetric monoidal categories, which is
compatible with the braiding $\sigma$ on both categories in that the following
diagram commutes for all $A,B\in\mathsf{C}$:

${{FA\otimes FB}}$${{FB\otimes FA}}$${{F(A\otimes B)}}$${{F(B\otimes
A)}}$$\scriptstyle{\sigma}$$\scriptstyle{F(\sigma)}$

######  Definition 3.2.2.3.

A _symmetric monoidal adjunction_ is a monoidal adjunction

${{\mathsf{C}}}$${{\mathsf{D}}}$$\scriptstyle{G}$$\scriptstyle{F}$$\scriptstyle{\dashv}$

between _symmetric_ monoidal categories $\mathsf{C},\mathsf{D}$ in which the
left adjoint $F$ is a strong _symmetric_ monoidal functor.

####  3.2.3. Symmetric monoidal monads: commutative monads

Adapting the notion of monad to the setting of symmetric monoidal categories
leads to the notion of _symmetric monoidal monad_. We will also use the term
_commutative monad_ as a synonym for “symmetric monoidal monad”. (Despite the
definition of “commutative monad” usually being slightly different, they are
equivalent concepts, see [Koc72].)

######  Definition 3.2.3.1 (Commutative Monad).

Let $(C,\boxtimes,I)$ be a symmetric monoidal category. A _symmetric monoidal
monad_ , or _commutative monad_ , $(T,i,m,\otimes)$ on $C$ is

  1. 1.

a monad $(T,i,m)$ on $C$, together with

  2. 2.

a natural transformation,

$\otimes^{X,Y}:TX\boxtimes TY\to T(X\boxtimes Y),\qquad(X,Y\in\mathsf{C})$

such that

  3. 3.

$(T,i,\otimes)$ is a monoidal functor, satisfying two further coherence
conditions which we omit (see [Sea12, 1.2] for the complete definition),

  4. 4.

such that the following diagram commutes, where $\sigma$ is the braiding (see
3.2.2.1):

${{TX\boxtimes TY}}$${{TY\boxtimes TX}}$${{T(X\boxtimes Y)}}$${{T(Y\boxtimes
X)}}$$\scriptstyle{\sigma}$$\scriptstyle{T\sigma}$$\scriptstyle{\otimes}$$\scriptstyle{\otimes}$

######  Example 3.2.3.2.

The free-$R$-module monad $R^{(-)}$ becomes a commutative monad on
$(\mathsf{Set},\times,1)$ if we let

$\otimes:R^{(X)}\times R^{(Y)}\to R^{(X\times
Y)},\;\;\;\;\;\;\;(X,Y\in\mathsf{Set})$

be the map defined by sending a pair of basis element $(x,y)\in R^{(X)}\times
R^{(Y)}$ to the basis element $(x,y)\in R^{(X\times Y)}$ and using the
universal property of free $R$-modules to extend by linearity.

######  Example 3.2.3.3.

The free-linear-$hk$-space becomes a commutative monad on
$(hk\mathsf{Top},\times,1)$, in the same way as the free-$R$-module monad,
_mutatis mutandis_ (using the universal property of free linear $hk$-spaces in
place of the one for free $R$-modules).

######  Remark 3.2.3.4.

One can generalise the previous two examples to the case of $R$-module objects
internal to a sufficiently well-behaved cartesian closed category, and even
further to the case of modules over _commutative algebraic theories_ in place
of $R$-modules, see [Low12].

##### Binary Morphisms and tensor products of modules over commutative monads.

The following is a generalisation of the notion of a bilinear map of modules
over a ring to the case of modules over a commutative monad:

######  Definition 3.2.3.5 (Binary morphism, [Sea12], 2.1).

Let $(T,i,m,\otimes)$ be a commutative monad over a symmetric monoidal
category $(\mathsf{C},\boxtimes,I)$. A _binary morphism_ of $T$-modules is a
$\mathsf{C}$-morphism $f:A\boxtimes B\to C$ (where $A,B,C$ are $T$-modules)
such that the following diagram commutes:

${{TA\boxtimes TB}}$${{T(A\boxtimes B)}}$${TC}$${{A\boxtimes
B}}$${C}$$\scriptstyle{\otimes}$$\scriptstyle{Tf}$$\scriptstyle{f}$$\scriptstyle{t^{A}\,\boxtimes\,t^{B}\;}$$\scriptstyle{t^{C}}$

######  Remark 3.2.3.6.

In [Sea12], binary morphisms of $T$-modules are referred to as “bimorphisms”.
This term is however slightly ambiguous, as it is also sometimes used for
morphisms which are both an epimorphism and a monomorphism.

The above definition of binary morphism may seem somewhat obscure. We can shed
some light on it by introducing a notion of “morphism in one variable” and
then showing that a binary morphism is nothing but a “morphism in _both_
variables”.

######  Definition 3.2.3.7 (Morphism in a variable).

Let $(T,i,m,\otimes)$ be a commutative monad over a symmetric monoidal
category $(\mathsf{C},\boxtimes,I)$, let $A,B,C$ be $T$-modules and let
$f:A\boxtimes B\to C$ be a $\mathsf{C}$-morphism.

  1. 1.

$f$ is a _morphism of $T$-modules in the left variable_ if, and only if, the
diagram

${{TA\boxtimes B}}$${{T(A\boxtimes B)}}$${TC}$${{A\boxtimes
B}}$${C}$$\scriptstyle{\otimes\circ(\mathsf{id}_{TA}\boxtimes
i^{B})}$$\scriptstyle{Tf}$$\scriptstyle{f}$$\scriptstyle{t^{A}\boxtimes\mathsf{id}_{B}\;}$$\scriptstyle{t^{C}}$

commutes.

  2. 2.

$f$ is a _morphism of $T$-modules in the right variable_ if, and only if, the
diagram

${{A\boxtimes TB}}$${{T(A\boxtimes B)}}$${TC}$${{A\boxtimes
B}}$${C}$$\scriptstyle{\otimes\circ(i^{A}\boxtimes\mathsf{id}_{TB})}$$\scriptstyle{Tf}$$\scriptstyle{t^{C}}$$\scriptstyle{f}$$\scriptstyle{\mathsf{id}_{A}\boxtimes
t^{B}\;}$

commutes.

If this definition is still not quite transparent, considering the case when
$\boxtimes$ is an ordinary cartesian product should clear it up:

######  Lemma 3.2.3.8.

Let $(T,i,m,\otimes)$ be a commutative monad over a concrete category
$\mathsf{C}$ with concrete finite products (i.e. $\mathsf{C}$ is equipped with
a “forgetful” faithful functor $\mathsf{C}\to\mathsf{Set}$, which we will
leave implicit, preserving finite products). Then a $\mathsf{C}$-morphism
$f:A\times B\to C$ is a morphism of $T$-modules in the left (right, resp.)
variable if, and only if, for any $b_{0}\in B$ (for any $a_{0}\in A$, resp.),
the map

$a\mapsto f(a,b_{0}),\;\;\;(b\mapsto f(a_{0},b),\text{ resp.})$

is a morphism of $T$-modules.

###### Proof.

Because of the symmetry of the situation, we consider only the case of a
morphism of $T$-modules in the left variable. For the “if” direction, suppose
that for all $b_{0}\in B$, $a\mapsto f(a,b_{0})$ is a morphism of $T$-modules.
Then for all $b_{0}\in B$, $\alpha\in TA$,

$\displaystyle f\circ(t^{A}\times\mathsf{id}_{B})(\alpha,b_{0})$
$\displaystyle=f(t^{A}(\alpha),b_{0})$
$\displaystyle=t^{C}(\,T(f(-,b_{0}))(\alpha)\,)$
$\displaystyle=t^{C}(\,T(f)(\alpha\otimes i^{B}(b_{0}))\,)$
$\displaystyle=t^{C}\circ Tf\circ\otimes\circ(\mathsf{id}_{TA}\times
i^{B})(\alpha,b_{0}).$

This shows that $f$ is a morphism of $T$-modules in the left variable.

For the “only if” direction, suppose that $f$ is a morphism of $T$-modules in
the left variable and let $b_{0}\in B$. Then

$\displaystyle f(t^{A}(\alpha),b_{0})$
$\displaystyle=f\circ(t^{A}\times\mathsf{id}_{B})(\alpha,b_{0})$
$\displaystyle=t^{C}\circ Tf\circ\otimes\circ(\mathsf{id}_{TA}\times
i^{B})(\alpha,b_{0})$ $\displaystyle=t^{C}(\,T(f)(\alpha\otimes
i^{B}(b_{0}))\,)$ $\displaystyle=t^{C}(\,T(f(-,b_{0}))(\alpha)\,)$

where we have used that $f$ is a morphism of $T$-modules in the left variable
in going from the first to the second line. This shows that $f(-,b_{0})$ is a
morphism of $T$-modules, completing the proof. ∎

We can now give a three equivalent characterisations of binary morphisms. The
second confirms the intuition that a binary morphism is indeed a “morphism in
both variables”, while the third hints on how to construct the _tensor product
of $T$-modules_.

######  Proposition 3.2.3.9 (Characterisations of binary morphisms).

As before, let $(T,i,m,\otimes)$ be a commutative monad over a monoidal
category $(\mathsf{C},\boxtimes,I)$, let $A,B,C$ be $T$-modules and let
$f:A\boxtimes B\to C$ be a $\mathsf{C}$-morphism. Then the following are
equivalent:

  1. 1.

$f$ is a binary morphism of $T$-modules.

  2. 2.

$f$ is both a morphism of $T$-modules in left variable and a morphism of
$T$-modules in the right variable.

  3. 3.

The diagram,

${{T(TA\boxtimes TB)}}$${{T(A\boxtimes B)}}$${C,}$$\scriptstyle{t^{T(A\times
B)}\circ\,{T(\otimes)}}$$\scriptstyle{T(t^{A}\boxtimes
t^{B})}$$\scriptstyle{t^{C}\circ\,Tf}$

commutes.

###### Proof.

The equivalence of the first two points is [Sea12, Proposition 2.1.2]; the
equivalence of the first and the last item is [Sea12, Lemma 2.3.2] ∎

##### The tensor product of $T$-modules

The following is a generalisation of the tensor product of modules over a
commutative ring to the setting of commutative monads.

######  Definition 3.2.3.10 (Tensor product of $T$-modules).

Let $(T,i,m,\otimes)$ be a commutative monad over a symmetric monoidal
category $(\mathsf{C},\boxtimes,I)$. The _tensor product_ $A\otimes_{T}B$ of
two $T$-modules $A$ and $B$ is defined, if it exists, as the co-equaliser (in
$\mathsf{M}$) of the two morphisms

${{T(TA\boxtimes TB)}}$${{T(A\boxtimes B)}.}$$\scriptstyle{T(\otimes)\circ
t^{\,TA\boxtimes TB}}$$\scriptstyle{T(t^{A}\boxtimes t^{B})}$

This construction satisfies the expected universal property [Sea12, Lemma
2.3.3]:

######  Proposition 3.2.3.11 (Universal property of tensor product).

As before, let $(T,i,m,\otimes)$ be a commutative monad over a symmetric
monoidal category $(\mathsf{C},\boxtimes,I)$ and let $A,B$ be objects of
$\mathsf{C}$ and suppose that the tensor product $A\otimes B$ exists. Then,
for every binary morphism of $T$-modules,

$f:A\boxtimes B\to C,$

to some object $C$ of $\mathsf{C}$, there is a unique morphism of $T$-modules,

$\overline{f}:A\otimes B\to C,$

such that the diagram,

${{A\otimes_{\mathsf{C}}B}}$${C}$${{A\boxtimes
B}}$$\scriptstyle{\overline{f}}$$\scriptstyle{\otimes_{\mathsf{C}}}$$\scriptstyle{f}$

commutes, where the map $A\boxtimes B\to A\otimes_{\mathsf{C}}B$ is the
composite of the maps

${{A\boxtimes B}}$${{T(A\boxtimes
B)}}$${{A\otimes_{M}B,}}$$\scriptstyle{i}$$\scriptstyle{q}$

with $q$ the canonical map to the coequaliser.

The category of modules over a symmetric monoidal monad is now itself
symmetric monoidal [Sea12, Theorem 2.2.5]:

######  Theorem 3.2.3.12.

Let $\mathsf{C}$ a symmetric monoidal category, let $T$ be a commutative monad
on $\mathsf{C}$ and suppose that the Eilenberg-Moore category $\mathsf{C}^{T}$
has all coequalisers. Then the tensor product of $T$-modules endows
$\mathsf{C}^{T}$ with a symmetric monoidal structure.

######  Example 3.2.3.13 (Tensor product of $R$-modules).

Let $R$ be a commutative ring and consider the category the free-$R$-module
monad $R^{(-)}$ (which is a commutative monad on $(\mathsf{Set},\times,1)$).
As discussed before, the category of $R^{(-)}$-modules is then equivalently
the category of $R$-modules. In this case, a binary morphism is precisely a
bilinear map (this is implied by 3.2.3.8) and then it follows from the
respective universal properties that the tensor product of $R^{(-)}$-modules
agrees with the usual tensor product.

######  Example 3.2.3.14 (Tensor product of linear $hk$-spaces).

Since linear $hk$-spaces are equivalently $F_{\mathbb{K}}$-modules (where
$F_{\mathbb{K}}$ is the free-linear-$hk$-space monad, a commutative monad),
3.2.3.10 yields a natural notion of tensor product of linear $hk$-spaces. We
denote this tensor product by $\otimes_{\mathbb{K}}$. By 3.2.3.12, we conclude
that the category ${\mathsf{Vect}(hk\mathsf{Top})}$ of linear $hk$-spaces
becomes a symmetric monoidal category under the tensor product
$\otimes_{\mathbb{K}}$.

As we will see, ${\mathsf{Vect}(hk\mathsf{Top})}$ is even a _closed symmetric
monoidal category_.

####  3.2.4. Closed symmetric monoidal categories

######  Definition 3.2.4.1.

A _closed symmetric monoidal category_ is a symmetric monoidal category
$\mathsf{C}$, equipped with an _internal hom_ functor,

$[-,-]_{\mathsf{C}}:\mathsf{C}^{\text{op}}\times\mathsf{C}\to\mathsf{C},$

and natural isomorphisms

$\tau^{A,B,C}:\operatorname{Hom}_{\mathsf{C}}(A\otimes_{\mathsf{C}}B,C)\cong\operatorname{Hom}_{\mathsf{C}}(A,[B,C]_{\mathsf{C}}),\qquad(A,C\in\mathsf{C})$

exhibiting $-\otimes B$ as left adjoint to $[B,-]_{\mathsf{C}}$, for every
object $B\in\mathsf{C}$. The adjunction $-\otimes B\dashv[B,-]_{\mathsf{C}}$
is called the _tensor-hom adjunction_. If unambiguous, we drop indices and
simply write $[-,-]$ instead of $[-,-]_{\mathsf{C}}$.

######  Remark 3.2.4.2.

Since adjoints are essentially unique, the internal hom of a closed symmetric
monoidal category is already essentially determined by its monoidal structure.
In light of this, a _closed structure_ on a symmetric monoidal category is
“property-like”.

######  Remark 3.2.4.3.

In [Bar91] (as well as other writings of Barr), closed symmetric monoidal
category are referred to “autonomous categories” (not to be confused with _$*$
-autonomous_ categories, which are a particular sort of symmetric monoidal
category, see 3.3.3.1).

######  Remark 3.2.4.4 (Relation to “external” hom functor).

Note that, by the tensor-hom adjunction,

$\operatorname{Hom}_{\mathsf{C}}(I,[-,-,])\cong\operatorname{Hom}_{\mathsf{C}}(I\otimes-,-)\cong\operatorname{Hom}_{\mathsf{C}}(-,-),$

where $I$ is the tensor unit. Hence, a “global element” $I\to[-,-]$ of the
internal hom corresponds bijectively to an element of the “external” hom
$\operatorname{Hom}_{\mathsf{C}}(-,-)$.

##### Examples of closed symmetric monoidal categories

When the tensor product is the cartesian product $\times$, the notion of
symmetric monoidal category reduces to that of a cartesian closed category. In
particular:

######  Example 3.2.4.5.

The categories of $k$-spaces, $hk$-spaces and QCB spaces are closed symmetric
monoidal categories under the cartesian product $\times$ as the tensor product
and the space $C(-,-)$ of continuous maps as the internal hom.

The most basic non-cartesian example is the following.

######  Example 3.2.4.6.

The category of $R$-modules over a commutative ring $R$ is a closed symmetric
monoidal category with respect to the tensor product of $R$-modules and the
usual internal hom.

More generally, commutative monads give rise to closed symmetric monoidal
categories [Sea12, Corollary 2.5.7]:

######  Theorem 3.2.4.7.

Let $\mathsf{C}$ be a closed symmetric monoidal category with all equalisers
and let $T$ be a commutative monad on $\mathsf{C}$ such that the Eilenberg-
Moore category $\mathsf{C}^{T}$ has all coequalisers. Then $\mathsf{C}^{T}$
becomes a closed symmetric monoidal category under the tensor product of
$T$-modules.

Most importantly to our cause, this implies that linear $hk$-spaces form a
closed symmetric monoidal category:

######  Example 3.2.4.8.

The category ${\mathsf{Vect}(hk\mathsf{Top})}$ of linear $hk$-spaces is
equivalently the category of $F_{\mathbb{K}}$-modules (with $F_{\mathbb{K}}$
the free-linear-$hk$-space monad) which satisfies the hypotheses of 3.2.4.7
(since $hk\mathsf{Top}$ is a complete category and
${\mathsf{Vect}(hk\mathsf{Top})}$ is cocomplete). Hence,
${\mathsf{Vect}(hk\mathsf{Top})}$ is a closed symmetric monoidal category
under the tensor product $\otimes_{\mathbb{K}}$ of $F_{\mathbb{K}}$-modules.
The internal hom that arises this way agrees with the space $L(-,-)$ from
2.2.2.2, as can be seen as follows. Let $V,W,Z$ be linear $hk$-spaces. By
cartesian closure of $hk\mathsf{Top}$, the set of continuous linear maps $V\to
L(W,Z)$ is in natural bijection with the set of continuous bilinear maps
$V\times W\to Z$. The latter can in turn be identified with the set of
continuous linear maps $V\otimes_{\mathbb{K}}W\to Z$. Hence,
$-\otimes_{\mathbb{K}}W$ is left adjoint to $L(W,-)$ and $L(-,-)$ is indeed
the internal hom associated to the tensor product $\otimes_{\mathbb{K}}$.

##### The internal tensor-hom adjunction

The tensor-hom adjunction “internalises” to the following statement.

######  Proposition 3.2.4.9 (Internal tensor-hom adjunction).

Let $\mathsf{C}$ be a closed symmetric monoidal category. Then we have a
natural isomorphism

$[A\otimes B,C]\cong[A,[B,C]].\;\;\;\;(A,B,C\in\mathsf{C})$

###### Proof.

Let $D$ be any object of $\mathsf{C}$. Applying the (“external”) tensor-hom
adjunction several times, we obtain the following chain of natural
isomorphisms.

$\displaystyle\operatorname{Hom}_{\mathsf{C}}(D,[A,[B,C]])$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(D,[A,[B,C]])$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(D\otimes A,[B,C])$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(D\otimes A\otimes B,C)$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(D,[A\otimes B,C]).$

The claim now follows from the Yoneda lemma. ∎

##### The hom-hom adjunction

Another adjunction arising from the tensor-hom adjunction, crucial for our
later considerations on duality theory, is the following.

######  Proposition 3.2.4.10 (Hom-hom adjunction).

Let $\mathsf{C}$ be a closed symmetric monoidal category. Then we have a
natural isomorphism

$\operatorname{Hom}_{\mathsf{C}}(A,[B,C])\cong\operatorname{Hom}_{\mathsf{C}^{\text{op}}}([A,C],B),\;\;\;\;(A,B,C\in\mathsf{C})$

giving an adjunction

${{\mathsf{C}^{\text{op}}}}$${{\mathsf{C}.}}$$\scriptstyle{[-,C]}$$\scriptstyle{[-,C]}$$\scriptstyle{\dashv}$

###### Proof.

Using the tensor-hom adjunction and symmetry of the tensor product,

$\displaystyle\operatorname{Hom}_{\mathsf{C}}(A,[B,C])$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(A\otimes B,C)$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(B\otimes A,C)$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(B,[A,C])$
$\displaystyle=\operatorname{Hom}_{\mathsf{C}^{\text{op}}}([A,C],B).$

∎

As an important corollary, we obtain:

######  Corollary 3.2.4.11.

Let $\mathsf{C}$ be a closed symmetric monoidal category. Then the tensor
product $\otimes$ preserves all colimits in both variables, and the internal
hom functor [-,-] preserves limits in the second variable and sends colimits
to limits in the first variable:

$\displaystyle(\operatorname*{colim}_{i}A_{i})\otimes(\operatorname*{colim}_{j}B_{j})$
$\displaystyle\cong\operatorname*{colim}_{i,j}(A_{i}\otimes B_{j}),$
$\displaystyle[\operatorname*{colim}_{i}A_{i},B]$
$\displaystyle\cong\lim_{i}[A_{i},B],$ $\displaystyle[A,\lim_{i}B_{i}]$
$\displaystyle\cong\lim_{i}[A,B_{i}].$

###### Proof.

Let $A$ be any object. Then the functors $A\otimes-$, $-\otimes A$ are left
adjoints and $[A,-]$ is a right adjoint (by the tensor-hom adjunction).
Furthermore, by the hom-hom adjunction, 3.2.4.10,

$[-,A]:\mathsf{C}^{\text{op}}\to\mathsf{C}$

is a left adjoint. The claim now follows from the fact that left adjoints
preserve colimits and right adjoints preserve limits. ∎

##### Day’s reflection theorem

Reflective subcategories of closed symmetric monoidal categories inherit their
closed monoidal structure under a number of equivalent simple conditions which
are summarised by the following theorem [Day72, Theorem 1.2]:

######  Theorem 3.2.4.12.

Let $\mathsf{C}$ be a closed symmetric monoidal category and suppose that

${{\mathsf{D}}}$${{\mathsf{C}}}$$\scriptstyle{L}$$\scriptstyle{\dashv}$

is a reflective subcategory with reflector $L$ (i.e. left adjoint to the
inclusion). Let

$\epsilon:A\to LA$

be the unit of the adjunction $(\hookrightarrow)\vdash L$ and consider the
following natural transformations.

$\displaystyle\epsilon^{[A,B]}$ $\displaystyle:[A,B]\to L[A,B],$
$\displaystyle(A\in\mathsf{C},\,B\in\mathsf{D})$
$\displaystyle[\epsilon^{A},B]$ $\displaystyle:[LA,B]\to[A,B],$
$\displaystyle(A\in\mathsf{C},\,B\in\mathsf{D})$ $\displaystyle
L(\epsilon\otimes B)$ $\displaystyle:L(A\otimes B)\to L(LA\otimes B),$
$\displaystyle(A,B\in\mathsf{D})$ $\displaystyle L(\epsilon\otimes\epsilon)$
$\displaystyle:L(A\otimes B)\to L(LA\otimes LB).$
$\displaystyle(A,B\in\mathsf{D})$

Then, if one of them is an isomorphism, so are the others. In this case,
$\mathsf{D}$ becomes a closed symmetric monoidal category with tensor product
$L(-\otimes-)$ and internal hom $[-,-]$, and $L$ is (or rather, admits the
structure of) a strong symmetric monoidal functor, thus making the adjunction
$(\hookrightarrow)\vdash L$ a symmetric monoidal adjunction.

###  3.3. Duality: Separated Objects, $*$-Autonomous Categories and the Chu
construction

We now discuss a general theory of duality in the setting of closed symmetric
monoidal categories which we will subsequently apply to linear $hk$-spaces.

##### Evaluation morphisms

First, we need a definition of a general version of evaluation mappings and
the canonical morphism $\eta:V\to V^{\wedge\wedge}$ to the double dual.

######  Definition 3.3.0.1.

Let $\mathsf{C}$ be a closed symmetric monoidal category and let $A,B,D$ be
objects of $\mathsf{C}$.

  1. 1.

The _evaluation morphism_ ,

$e^{A,B}:[A,B]\otimes A\to B,$

is given by

$e^{A,B}:=(\tau^{[A,B],A,B})^{-1}(\mathsf{id}_{[A,B]}).$

  2. 2.

The _canonical morphism to the double dual of $A$ with respect to $D$_ is the
morphism

$\eta^{A,D}:A\to[[A,D],D]$

defined as

$\eta^{A,D}:=\tau^{A,[A,D],D}(e^{A,D}\circ\sigma^{A,[A,D]}),$

where $\sigma$ is the braiding (see 3.2.2.1).

Examining the proof of the hom-hom adjunction (3.2.4.10), we see that
$\eta^{-,D}$ is precisely the unit of this adjunction. In particular,

$\eta^{-,D}:\mathsf{id}_{\mathsf{C}}\to[[-,D],D]$

is a natural transformation.

####  3.3.1. Regular monomorphisms, regular images and (co)regular categories

At this point, we would like to ask whether $\eta^{A,D}$ is – in a suitable
sense – an embedding, and what its image is. This would then yield a general
formulation of what it means for the dual $[A,D]$ to “separate the points” of
$A$ and enable the study of $A$ in terms of its dual $[A,D]$. The category-
theoretic notion of embedding which we would like to adhere to is that of
_regular monomorphism_ with the corresponding notion of image being the
_regular image_. Categories in which these notions are particularly well-
behaved are known as _coregular categories_ (the dual notion of which is that
of a _regular category_).

##### Regular mono- and epimorphisms

We begin with the notions of regular mono- and epimorphism.

######  Definition 3.3.1.1.

Let $\mathsf{C}$ be a category. A _regular monomorphism_ is a morphism which
is the equaliser of some pair of morphisms. A _regular epimorphism_ in
$\mathsf{C}$ is a regular monomorphism in the opposite category
$\mathsf{C}^{\text{op}}$, i.e. a morphism (in $\mathsf{C}$) which is the _co_
equaliser of some pair of morphisms.

As the terminology suggests:

######  Proposition 3.3.1.2.

A regular monomorphism is always a monomorphism.

###### Proof.

By definition, a regular monomorphism $m:A\to B$ is the equaliser of a pair of
maps $k,l:B\to C$. For any two morphisms $f,g:D\to A$ with $m\circ f=m\circ
g$, the following diagram commutes:

${A}$${B}$${C}$${D}$$\scriptstyle{m}$$\scriptstyle{k}$$\scriptstyle{l}$$\scriptstyle{m\circ
g}$$\scriptstyle{m\circ f}$

Hence, by the universal property of the equaliser, there is a _unique_
morphism $u$ making the following diagram commute.

${A}$${B}$${C}$${D}$$\scriptstyle{m}$$\scriptstyle{k}$$\scriptstyle{l}$$\scriptstyle{m\circ
g}$$\scriptstyle{m\circ f}$$\scriptstyle{u}$

Therefore, $f=u=g$. Since $f,g$ were arbitrary morphisms with $m\circ f=m\circ
g$, this shows that $m$ is a monomorphism. ∎

Recall that a _split epimorphism_ is a morphism which admits a section (i.e. a
right inverse). Dually, a _split monomorphism_ is a morphism which admits a
retraction (i.e. a left inverse).

######  Lemma 3.3.1.3.

Let $\mathsf{C}$ be a category with equalisers. Then every split epimorphism
(split monomorphism, resp.) is a regular epimorphism (monomorphism, resp.).

###### Proof.

If $i:A\to B$ is is a split monomorphism and $p$ is a left inverse of $i$,
then $i$ is the equaliser of $i\circ p$ and $\mathsf{id}_{B}$ and hence a
regular monomorphism. The case of split epimorphisms is dual. ∎

##### Characterising (regular, split) mono- and epimorphisms in
$hk\mathsf{Top}$

In order to be able to apply this general discussion to the examples that we
are interested in, we also need to characterise the regular mono- and
epimorphisms in the category of $hk$-spaces, where they correspond to
important notions of general topology. The following can be found in [Str09,
Theorem 3.1]:

######  Theorem 3.3.1.4 (Morphisms in $hk\mathsf{Top}$).

Let $f:X\to Y$ be a continuous map of $hk$-spaces. Then we have the following
equivalences:

$\displaystyle f\text{ is a monomorphism }$
$\displaystyle\Leftrightarrow\;f\text{ is injective, }$ $\displaystyle f\text{
is a regular monomorphism }$ $\displaystyle\Leftrightarrow\;f\text{ is a
closed embedding, }$ $\displaystyle f\text{ is a split monomorphism }$
$\displaystyle\Leftrightarrow\;\text{$X$ is a retract of $Y$ and
$f:X\hookrightarrow Y$ is the inclusion, }$

and dually,

$\displaystyle f\text{ is an epimorphism }$
$\displaystyle\Leftrightarrow\;f\text{ has dense image, }$ $\displaystyle
f\text{ is a regular epimorphism }$ $\displaystyle\Leftrightarrow\;f\text{ is
a quotient map, }$ $\displaystyle f\text{ is a split epimorphism }$
$\displaystyle\Leftrightarrow\;\text{ $Y$ is a retract of $X$ and
$f:X\twoheadrightarrow Y$ is the retraction. }$

Moreover, we have the following permanence properties, where all limits and
colimits are taken in $hk\mathsf{Top}$:

  1. 1.

Arbitrary products, coproducts, pushouts or composites of regular
monomorphisms (i.e. closed embedding, by the above) are regular monomorphisms.

  2. 2.

Arbitrary coproducts, _finite_ products, pullbacks and composites of regular
epimorphisms (i.e quotient maps, by the above) are regular epimorphisms.

Using this characterisation, we may similarly characterise the regular
monomorphisms in ${\mathsf{Vect}(hk\mathsf{Top})}$, which we will need in
order to apply our general duality theoretic results to the case of linear
$hk$-spaces.

######  Corollary 3.3.1.5.

Let $V,W$ be linear $hk$-spaces and let $i:V\to W$ be a continuous linear map.
Then $i$ is a regular monomorphism if, and only if, it is a regular
monomorphism in $hk\mathsf{Top}$, i.e. a closed embedding.

###### Proof.

For the “if” direction, note that if $i$ is a closed embedding, then it is the
equaliser of the projection

$p:W\to W/i(V)$

and the zero map $0:W\to W/i(V)$.

For the other direction, recall that the forgetful functor
${\mathsf{Vect}(hk\mathsf{Top})}\to hk\mathsf{Top}$ is a right adjoint (to the
free linear $hk$-space functor $F_{\mathbb{K}}$) and that right adjoints
preserve limits. Hence, if $i$ is a regular monomorphism in
${\mathsf{Vect}(hk\mathsf{Top})}$, it is also a regular monomorphism in
$hk\mathsf{Top}$. ∎

##### The regular image

######  Definition 3.3.1.6.

Let $\mathsf{C}$ be a category and let $f:A\to B$ be a morphism.

  1. 1.

The _regular image_ $\operatorname*{im}(f)$ of $f$ is, if it exists, the
equaliser of the cokernel pair of $f$, i.e. the equaliser of the pair of
morphisms $B\to B\sqcup_{f}B$ arising from the pushout diagram:

${A}$${B}$${B}$${{B\sqcup_{f}B}}$$\scriptstyle{f}$$\scriptstyle{f}$

  2. 2.

Dually, the _regular coimage_ $\operatorname*{coim}(f)$ of $f$ is, if it
exists, the coequaliser of the kernel pair of $f$. Put differently, the
regular coimage of $f$ is the image of $f$ considered as an morphism in the
opposite category $\mathsf{C}$.

  3. 3.

The universal property of the pushout yields a unique morphism,

$f|^{\operatorname*{im}(f)}:A\to\operatorname*{im}(f),$

the _corestriction of $f$ to its image_, making the following diagram commute:

${A}$${{\operatorname*{im}(f)}}$${B}$${B}$${{B\sqcup_{f}B}}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{f|^{\operatorname*{im}(f)}}$

  4. 4.

The factorisation

${A}$${{\operatorname*{im}(f)}}$${B}$$\scriptstyle{f|^{\operatorname*{im}(f)}}$

is called the _regular image factorisation_ of $f$.

##### Regular and coregular categories

######  Definition 3.3.1.7 ((Co)regular category, [Gra21]).

A category $\mathsf{C}$ is _coregular_ if

  1. 1.

it has all finite colimits,

  2. 2.

the regular image of every morphism in $\mathsf{C}$ exists,

  3. 3.

and regular monomorphisms are stable under pushouts.

$\mathsf{C}$ is _regular_ if $\mathsf{C}^{\text{op}}$ is coregular, and
_biregular_ if it is both regular and coregular.

The following shows that in coregular categories, the regular image behaves as
one would hope [Gra21, Theorem 1.11]:

######  Theorem 3.3.1.8.

Let $\mathsf{C}$ be a coregular category let $f$ be a morphism. Then:

  1. 1.

The corestriction of $f$ to its image

$f|^{\operatorname*{im}(f)}:A\to\operatorname*{im}(f)$

is an epimorphism. Hence, the regular image factorisation factors $f$ into an
epimorphism followed by a regular monomorphism:

${A}$${{\operatorname*{im}(f)}}$${B}$$\scriptstyle{f|^{\operatorname*{im}(f)}}$

  2. 2.

Such factorisation is essentially unique in that for any further epimorphism
$e$ and regular monomorphism $m$, there is a unique isomorphism
$i:I\to\operatorname*{im}(f)$ making the following diagram commute

${{\operatorname*{im}(f)}}$${A}$${B}$${I}$$\scriptstyle{f|^{\operatorname*{im}(f)}}$$\scriptstyle{e}$$\scriptstyle{m}$$\scriptstyle{i}$

In fact, coregular categories are _precisely_ those categories in which every
morphism can be factored into an epimorphism followed by a regular
monomorphism in a well-behaved manner (see [Gra21, Theorem 1.14]):

######  Proposition 3.3.1.9.

Let $\mathsf{C}$ be a category with finite colimits in which regular
monomorphisms are stable under pullback. Then $\mathsf{C}$ is coregular if,
and only if, every morphism admits a factorisation into an epimorphism
followed by regular monomorphism.

##### The category of $hk$-spaces is biregular

Cartesian closure is not the only convenience that $hk$-spaces provide. As
shown in [Str09, Theorem 3.1 (f)]:

######  Proposition 3.3.1.10.

The category $hk\mathsf{Top}$ of $hk$-spaces is biregular, i.e. both a
coregular category and a regular category.

##### Properties of regular monomorphisms in coregular categories

As a final note on (co)regular categories, let us record the following fact
(see [Gra21, Proposition 1.13, Lemma 1.3] for a proof).

######  Proposition 3.3.1.11.

Let $\mathsf{C}$ be a coregular category and let $f,g$ be morphisms.

  1. 1.

The composite $f\circ g$ of any pair of regular monomorphisms $f$, $g$ is a
regular monomorphisms, as well.

  2. 2.

If $g\circ f$ is a regular monomorphism, then $f$ is also a regular
monomorphism.

  3. 3.

If $f$ is both a regular monomorphism and an epimorphism, then $f$ is an
isomorphism.

####  3.3.2. The triple dualisation lemma and separated objects

##### The triple dualisation lemma

The following simple observation will be crucial.

######  Lemma 3.3.2.1 (triple dualisation lemma).

Let $\mathsf{C}$ be a closed symmetric monoidal category and let $D$ be any
object of $\mathsf{C}$. Then

$[\eta^{C,D},D]\circ\eta^{[C,D],D}=\mathsf{id}_{[C,D]},$

i.e. the canonical morphism to the “triple dual” with respect to $D$ (see
3.3.0.1),

$\eta^{[C,D],D}:[C,D]\to[[[C,D],D],D],$

has a right inverse, given by

$[\eta^{C,D},D]:[[[C,D],D],D]\to[C,D].$

In particular, $\eta^{[C,D],D}$ is a split (and hence regular) monomorphism.

###### Proof.

An adjunction can be characterised by the _triangle identities_ (see [Bor94,
Theorem 3.1.5], or any other category theory textbook). The triangle
identities for the hom-hom adjunction (see 3.2.4.10) reduce to a single
identity (because of the “self-adjoint” nature of this adjunction), and this
identity is given precisely by:

$[\eta^{C,D},D]\circ\eta^{[C,D],D}=\mathsf{id}_{[C,D]}.$

∎

##### Separated objects in a closed symmetric monoidal category

The triple dualisation lemma suggests the following definition.

######  Definition 3.3.2.2.

Let $\mathsf{C}$ be a closed symmetric monoidal category and let $D$ be an
object. An object $A$ of $\mathsf{C}$ is _$D$ -separated_ if the canonical map
(see 3.3.0.1).

$\eta^{A,D}:A\to[[A,D],D]$

is a regular monomorphism.

######  Example 3.3.2.3.

Let $K$ be a field. Then every (algebraic) $K$-vector space is $K$-separated
(assuming the axiom of choice).

The example that will be most important to our context is the following.

######  Example 3.3.2.4.

The $\mathbb{K}$-separated objects in the closed symmetric monoidal category
${\mathsf{Vect}(hk\mathsf{Top})}$ of linear $hk$-spaces are precisely the
_replete_ linear $hk$-spaces. Recall that a replete linear $hk$-space is a
linear $hk$-space for which the canonical map $V\to V^{\wedge\wedge}$ is a
closed embedding (1.4.2.1). Since regular monomorphisms in
${\mathsf{Vect}(hk\mathsf{Top})}$ are exactly given by closed embeddings
(3.3.1.5), this is exactly the definition of a $\mathbb{K}$-separated object
in ${\mathsf{Vect}(hk\mathsf{Top})}$.

######  Remark 3.3.2.5.

The notion of $D$-separated object can be seen as a special case of a
construction related to the _(enriched) idempotent core of a monad_ , see
[LW14, Example 4.18].

##### Some permanence properties of separated objects

It turns out that the category of $D$-separated objects enjoys excellent
permanence properties, thanks to the triple dualisation lemma. The next
proposition is a first step towards establishing this.

######  Proposition 3.3.2.6.

Let $\mathsf{C}$ be a coregular closed symmetric monoidal category and let $D$
be an object. Then:

  1. 1.

For any object $A$, $[A,D]$ is $D$-separated.

  2. 2.

If $B$ is $D$-separated and $\iota:A\hookrightarrow B$ is a regular
monomorphism, then $A$ is $D$-separated, as well. In particular, an object $A$
is $D$-separated if, and only if, it admits a regular monomorphism into _some_
dual $[B,D]$ with respect to $D$.

  3. 3.

For all objects $A,B\in\mathsf{C}$, if $B$ is $D$-separated, then $[A,B]$ is
also $D$-separated.

###### Proof.

  1. 1.

This follows directly from the triple dualisation lemma (3.3.2.1).

  2. 2.

By naturality of $\eta$, the following diagram commutes.

${B}$${{[[B,D],D]}}$${A}$${{[[A,D],D]}}$$\scriptstyle{\eta^{B,D}}$$\scriptstyle{\iota}$$\scriptstyle{\eta^{A,D}}$$\scriptstyle{[[\iota,D],D]}$

By 3.3.1.11, the composition of regular monomorphisms is a regular
monomorphism and hence,

$h:=\eta^{B,D}\circ\iota$

is a regular monomorphism. Since $h=[[\iota,D],D]\circ\eta^{A,D}$ (by
commutativity of the above diagram), $\eta^{A,D}$ is a regular monomorphism as
well, by 3.3.1.11. This shows that $A$ is $D$-separated.

  3. 3.

Since $B$ is $D$-separated, $\eta^{B,D}$ is a regular monomorphism. Since the
internal hom preserves limits in the second variable,

$[A,\eta^{B,D}]:[A,B]\to[A,[[B,D],D]]$

is also a regular monomorphism. Hence, $[A,B]$ admits a regular monomorphism
into a $D$-separated object and is therefore $D$-separated, by the previous
point.

∎

##### The separation functor

As we will see, $D$-separated objects form a reflective subcategory of the
given category (if the latter is sufficiently well-behaved). Let us first
define the would-be reflector, the _$D$ -separation_.

######  Definition 3.3.2.7 ($D$-Separation).

Let $\mathsf{C}$ be a coregular closed symmetric monoidal category and let $D$
be an object. We define the _$D$ -separation functor_ as the regular image
(see 3.3.1.6),

$\mathsf{sep}_{D}:=\operatorname*{im}(\eta^{-,D}),$

of the natural transformation $\eta^{-,D}$ in the functor category
$\mathsf{C}^{\mathsf{C}}$.

Since limits and colimits in functor categories are computed pointwise (see,
for example, [Rie17, Proposition 3.3.9]), the $D$-separation
$\mathsf{sep}_{D}(A)$ of an _object_ $A$ is the regular image in $\mathsf{C}$
of the canonical morphism to the double dual of $A$ with respect to $D$. The
regular image factorisation of $\eta^{-,D}$ furthermore yields two natural
transformations,

$A\to\mathsf{sep}_{D}(A)\hookrightarrow[[A,D],D],\;\;\;\;(A\in\mathsf{C}$

the corestriction of $\eta^{A,D}$ to its regular image, and the inclusion of
$\mathsf{sep}_{D}(A)$ into $[[A,D],D]$.

Of course, the term “$D$-separation” is only justified if the $D$-separation
of an object is $D$-separated. This is indeed the case.

######  Proposition 3.3.2.8.

Let $\mathsf{C}$ be a coregular closed symmetric monoidal category and let
$A,D$ be objects of $\mathsf{C}$. Then:

  1. 1.

$\mathsf{sep}_{D}(A)$ is always $D$-separated.

  2. 2.

The following are equivalent:

    1. 2.1.

$A$ is $D$-separated.

    2. 2.2.

The co-restriction of $\eta^{A,D}$ to its regular image,

$\eta^{A,D}|^{\mathsf{sep}_{D}(A)}:A\to\mathsf{sep}_{D}(A),$

is an isomorphism.

    3. 2.3.

There is _some_ isomorphism

$A\cong\mathsf{sep}_{D}(A).$

In particular, we can use the second point to reformulate the first as saying:

$\mathsf{sep}_{D}(\mathsf{sep}_{D}(A))\cong\mathsf{sep}_{D}(A).$

###### Proof.

  1. 1.

$\mathsf{sep}_{D}(A)$ admits a regular monomorphism into a dual with respect
to $D$, namely the inclusion

$\mathsf{sep}_{D}(A)\hookrightarrow[[A,D],D].$

Hence, the claim follows from 3.3.2.6.

  2. 2.

If $A$ is $D$-separated, i.e. $\eta^{A,D}$ is a regular monomorphism, then
$\eta^{A,D}|^{\mathsf{sep}_{D}(A)}$ is a regular monomorphism, as well (by
3.3.1.11). Since $\eta^{A,D}|^{\mathsf{sep}_{D}(A)}$ is also an epimorphism,
it is hence an isomorphism (by 3.3.1.11). On the other hand, if
$\eta^{A,D}|^{\mathsf{sep}_{D}(A)}$ is an isomorphism, then $\eta^{A,D}$ is a
regular monomorphism, so $A$ is $D$-separated.

The remaining implications follow from the fact that $\mathsf{sep}_{D}(A)$ is
always D-separated (this was the first point) and that the property of being
$D$-separated is isomorphism-invariant.

∎

This also shows that the essential image of the $D$-separation functor
$\mathsf{sep}_{D}$ is given precisely be the $D$-separated objects. The full
subcategory of a coregular closed symmetric monoidal category $\mathsf{C}$
spanned by the $D$-separated objects is therefore appropriately denoted by

$\mathsf{sep}_{D}\mathsf{C}\hookrightarrow\mathsf{C}.$

######  Definition 3.3.2.9.

For the case of replete linear $hk$-spaces and $D=\mathbb{K}$ (see Example
3.3.2.4), we write

$r:=\mathsf{sep}_{\mathbb{K}}$

for the $\mathbb{K}$-separation functor and call this functor the _repletion_.
Accordingly, the category of replete linear $hk$-spaces will be denoted by

$r{\mathsf{Vect}(hk\mathsf{Top})}:=\mathsf{sep}_{\mathbb{K}}{\mathsf{Vect}(hk\mathsf{Top})}.$

Unfolding definitions, the repletion $rV$ of a linear $hk$-space is precisely
the closure of its image in the double dual $V^{\wedge\wedge}$ under the
canonical map $V\to V^{\wedge\wedge}$.

#####  $D$-separated objects form a reflective subcategory

By now, the reader might expect, as the name would suggest, that the
$D$-separation is the reflector associated to the _reflective_ subcategory of
$D$-separated objects. That this expectation is indeed met is the content of
the following theorem.

######  Theorem 3.3.2.10.

Let $\mathsf{C}$ be a coregular closed symmetric monoidal category and let $D$
be an object. Then the $D$-separation functor (co-restricted to its image),

$\mathsf{sep}_{D}:\mathsf{C}\to\mathsf{sep}_{D}\mathsf{C},$

is left adjoint to the inclusion

$\mathsf{sep}_{D}\mathsf{C}\hookrightarrow\mathsf{C}.$

###### Proof.

In order to simplify notation, write

$(-,-):=\operatorname{Hom}_{\mathsf{C}}(-,-),$ $\eta^{A}:=\eta^{A,D},$

and,

$\mathsf{sep}:=\mathsf{sep}_{D},\;\;j^{A}:=\eta^{A}|^{\mathsf{sep}A}.$

In order to obtain an adjunction,

$(\mathsf{sep}A,B)\cong(A,B),\;\;\;\;\;\;(A,B\in\mathsf{C})$

define two natural transformations $\Phi$, $\Psi$ with components

$\Phi^{A,B}:(\mathsf{sep}A,B)\to(A,B),\;\;\Phi^{A,B}:=(j^{A},B)$
$\Psi^{A,B}:(A,B)\to(\mathsf{sep}A,B),\;\;\Psi^{A,B}:=(\mathsf{sep}(A),(j^{B})^{-1})\circ\mathsf{sep}^{A,B},$

where

$\mathsf{sep}^{A,B}:(A,B)\to(\mathsf{sep}(A),\mathsf{sep}(B)),\;\;f\mapsto\mathsf{sep}(f)$

is the action of the $D$-separation functor on morphisms $A\to B$. Fix two
objects $A,B$ and write $\Phi:=\Phi^{A,B}$, $\Psi:=\Psi^{A,B}$. We claim that
$\Phi$ and $\Psi$ are inverse to one another, which would complete the proof.
We will show this in two steps.

  1. 1.

First, we claim that

$j^{\mathsf{sep}(A)}=\mathsf{sep}(j^{A}).$

This can be seen as follows. By naturality of $j$, the diagram,

${A}$${{\mathsf{sep}(A)}}$${{\mathsf{sep}(A)}}$${{\mathsf{sep}(\mathsf{sep}(A))}}$$\scriptstyle{j^{A}}$$\scriptstyle{j^{A}}$$\scriptstyle{\mathsf{sep}(j^{A})}$$\scriptstyle{j^{\mathsf{sep}(A)}}$

commutes, i.e.

$j^{\mathsf{sep}(A)}\circ j^{A}=\mathsf{sep}(j^{A})\circ j^{A}.$

Since $j^{A}$ is an epimorphism (by 3.3.1.8), this implies the (intermediate)
claim.

  2. 2.

Now, using naturality of $j$, we find that

$\displaystyle\Phi\circ\Psi(f)$
$\displaystyle=(j^{A},B)\circ(\mathsf{sep}(A),(j^{B})^{-1})\circ\mathsf{sep}^{A,B}(f)$
$\displaystyle=(j^{A},B)((\mathsf{sep}(A),(j^{B})^{-1})(\mathsf{sep}^{A,B}(f))$
$\displaystyle=(j^{A},B)((j^{B})^{-1}\circ\mathsf{sep}^{A,B}(f))$
$\displaystyle=(j^{B})^{-1}\circ\mathsf{sep}^{A,B}(f)\circ j^{A}$
$\displaystyle=f,$

for all $f\in(A,B)$. Likewise, for every $g\in(\mathsf{sep}A,B)$,

$\displaystyle\Psi\circ\Phi$
$\displaystyle=(\mathsf{sep}(A),(j^{B})^{-1})\circ\mathsf{sep}^{A,B}\circ(j^{A},B)(g)$
$\displaystyle=(\mathsf{sep}(A),(j^{B})^{-1})(\mathsf{sep}((j^{A},B)(g)))$
$\displaystyle=(j^{B})^{-1}\circ\mathsf{sep}(g\circ j^{A}))$
$\displaystyle=(j^{B})^{-1}\circ\mathsf{sep}(g)\circ\mathsf{sep}(j^{A})$ (by
functoriality of $\mathsf{sep}$)
$\displaystyle=(j^{B})^{-1}\circ\mathsf{sep}(g)\circ j^{\mathsf{sep}(A)}$ (by
the previous step) $\displaystyle=g.$ (by naturality of $j$)

This completes the proof. ∎

The adjunction from 3.3.2.10 can be enhanced to an internal (or _enriched_)
version.

######  Corollary 3.3.2.11.

Let $\mathsf{C}$ be a coregular closed symmetric monoidal category and let $D$
be an object of $\mathsf{C}$. Then we have a natural isomorphism

$\displaystyle[A,B]\cong[\mathsf{sep}_{D}(A),B].\;\;\;\;\;\;\;\;(A\in\mathsf{C},B\in\mathsf{sep}_{D}\mathsf{C})$

###### Proof.

Let $C$ be any object. By 3.3.2.6, for any $D$-separated object $B$, $[C,B]$
is $D$-separated, as well. Combining this fact with 3.3.2.10, we get the
following chain of natural isomorphisms.

$\displaystyle\operatorname{Hom}_{\mathsf{C}}(C,[\mathsf{sep}(A),B])$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(\mathsf{sep}(A),[C,B])$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(A,[C,B])$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(C,[A,B]).$

The claim now follows from the Yoneda lemma. ∎

#####  $D$-separated objects form a closed symmetric monoidal category

Day’s reflection theorem (3.2.4.12) now immediately yields:

######  Corollary 3.3.2.12.

Let $\mathsf{C}$ be a coregular closed symmetric monoidal category and let $D$
be any object. Then the category $\mathsf{sep}_{D}(\mathsf{C})$ of
$D$-separated objects is a closed symmetric monoidal category with tensor
product $\mathsf{sep}_{D}(-\otimes_{\mathsf{C}}-)$ and internal hom
$[-,-]_{\mathsf{C}}$, and the $D$-separation functor is a strong symmetric
monoidal functor

$\mathsf{sep}_{D}:\mathsf{C}\to\mathsf{sep}_{D}(\mathsf{C}).$

Applying this Corollary to the case of replete linear $hk$-spaces, we obtain:

######  Corollary 3.3.2.13.

The category $r{\mathsf{Vect}(hk\mathsf{Top})}$ of replete linear $hk$-spaces
becomes a closed symmetric monoidal category under the internal hom $L(-,-)$
and tensor product

$V\,\otimes_{r}W:=r(V\otimes W),\qquad(V,W\in
r{\mathsf{Vect}(hk\mathsf{Top})})$

where $\otimes$ denotes the tensor product of linear $hk$-spaces
(equivalently, $F_{\mathbb{K}}$-modules, see Example 3.2.4.8) and
$r=\mathsf{sep}_{\mathbb{K}}$ is the repletion functor (3.3.2.9), i.e. the
$\mathbb{K}$-separation functor in the category of linear $hk$-spaces.

####  3.3.3. $*$-Autonomous categories

######  Definition 3.3.3.1.

A _$*$ -autonomous category_ is a closed symmetric monoidal category
$\mathsf{C}$ together with an object $D$, the _dualising object_ , such that

$\eta^{A,D}:A\to[[A,D],D],$

is an isomorphism for all $A\in\mathsf{C}$. In this case, we denote by
$(-)^{*}:=[-,D]$ the “dualisation functor”,

$(-)^{*}:\mathsf{C}^{\text{op}}\to\mathsf{C},$

with respect to $D$, which is then an equivalence of categories.

Hence, $*$-autonomous categories are “self-dual”,

$\mathsf{C}^{\text{op}}\cong\mathsf{C},$

a property with far-reaching consequences. For example, a $*$-autonomous
category $\mathsf{C}$ which has all limits automatically admits all colimits
as well (and vice versa). After all, colimits in $\mathsf{C}$ are just limits
in the opposite category $\mathsf{C}^{\text{op}}$ (which is equivalent to
$\mathsf{C}$ by $*$-autonomy). Of course, self-duality entails more generally
that for _any_ category-theoretic property of $\mathsf{C}$ (in the sense of
being invariant under equivalence of categories), its dual property is also
true of $\mathsf{C}$.

######  Example 3.3.3.2.

The category of finite-dimensional vector spaces over a field $K$ is a
$*$-autonomous category with respect to the standard tensor product $\otimes$
and internal hom, with $K$ as the dualising object.

####  3.3.4. The Chu construction

A general method to construct $*$-autonomous categories from closed symmetric
monoidal categories is given by the _Chu construction_ (given originally in
[Chu78]).

######  Definition 3.3.4.1 (Chu construction, [Bar91]).

Let $\mathsf{C}$ be a closed symmetric monoidal category and let $D$ be an
object. Then the category

$\mathsf{Chu}_{D}(\mathsf{C})$

is defined as follows.

  1. 1.

An object of $\mathsf{Chu}_{D}(\mathsf{C})$ is a triple $(A,A^{\top},a)$,
where

    1. 1.1.

$A,A^{\top}$ are objects of $\mathsf{C}$, and

    2. 1.2.

$a:A\otimes_{\mathsf{C}}A^{\top}\to D$ is a morphism in $\mathsf{C}$.

  2. 2.

A morphism $(A,A^{\top},a)\to(B,B^{\top},b)$ in $\mathsf{Chu}_{D}(\mathsf{C})$
is a pair $(f,f^{\top})$, where

    1. 2.1.

$f:A\to B$, $f^{\top}:B^{\top}\to A^{\top}$ are morphisms in $\mathsf{C}$,

    2. 2.2.

making the following diagram commute:

${{A\otimes_{\mathsf{C}}B^{\top}}}$${{A\otimes_{\mathsf{C}}A^{\top}}}$${{B\otimes_{\mathsf{C}}B^{\top}}}$${D}$$\scriptstyle{f\otimes
B^{\top}}$$\scriptstyle{A\otimes f^{\top}}$

To understand this definition, it is helpful to consider the case when
$\mathsf{C}$ is the category of $\mathbb{K}$-vector spaces and the dualising
object is $\mathbb{K}$. In this case, an object of
$\mathsf{Chu}_{\mathbb{K}}(\mathsf{Vect})$ is a pair of vector spaces equipped
with a bilinear pairing $\langle-,-\rangle$ to the base field $\mathbb{K}$,
and a morphism,

$(V,V^{\top},\langle-,-\rangle_{V})\to(W,W^{\top},\langle-,-\rangle_{W}),$

is a pair of linear maps $(f,f^{\top})$ that are “adjoint” to one another,

$\langle f(v),w^{\top}\rangle_{W}=\langle
v,f(w^{\top})\rangle_{V}.\;\;\;\;\;\;\;(v\in V,w^{\top}\in W^{\top})$

Let us now describe the dualising object, internal hom and tensor product on
${\mathsf{Chu}}_{D}(\mathsf{C})$.

######  Definition 3.3.4.2.

Let $\mathsf{C}$ be a closed symmetric monoidal category with pullbacks and
let $\mathcal{A}=(A,A^{\top},a)$, $\mathcal{B}=(B,B^{\top},b)$ be two objects
of $\mathsf{Chu}_{D}(\mathsf{C})$.

  1. 1.

Define dualising object of ${\mathsf{Chu}}_{D}(\mathsf{C})$ as

$\mathcal{D}:=(D,I_{\mathsf{C}},\rho),$

where $I_{\mathsf{C}}$ is the tensor unit and

$\rho:D\otimes_{\mathsf{C}}I_{\mathsf{C}}\to D$

is the right unitor.

  2. 2.

The internal hom

$[\mathcal{A},\mathcal{B}]_{{\mathsf{Chu}}_{D}(\mathsf{C})}$

is given by the triple $([A,B]_{0},A\otimes_{\mathsf{C}}B^{\top},c)$, where:

    1. 2.1.

$[A,B]_{0}$ is given by the pullback,

${{[A,B]_{0}}}$${{[B^{\top},A^{\top}]_{\mathsf{C}}}}$${{[A,B]_{\mathsf{C}}}}$${{[A\otimes
B^{\top},D]_{\mathsf{C}}}}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$

Here, $\phi$ is obtained via the tensor-hom adjunction from the composite,

${{A\otimes_{\mathsf{C}}B^{\top}\otimes_{\mathsf{C}}[B^{\top},A^{\top}]_{\mathsf{C}}}}$${{A\otimes_{\mathsf{C}}A^{\top}}}$${D}$$\scriptstyle{\mathsf{id}_{A}\otimes_{\mathsf{C}}\mathsf{eval}}$$\scriptstyle{a}$

and, similarly, $\psi$ is obtained from the composite,

${{[A,B]_{\mathsf{C}}\otimes_{\mathsf{C}}A\otimes_{\mathsf{C}}B^{\top}}}$${{B\otimes_{\mathsf{C}}B^{\top}}}$${D.}$$\scriptstyle{\mathsf{eval}\otimes_{\mathsf{C}}\mathsf{id}_{B^{\top}}}$$\scriptstyle{b}$

    2. 2.2.

The definition of $[A,B]_{0}$ yields a map

$[A,B]_{0}\to[A\otimes_{\mathsf{C}}B^{\top},D],$

which induces the pairing

$c:[A,B]_{0}\otimes(A\otimes_{\mathsf{C}}B^{\top})\to D$

via the tensor-hom adjunction.

  3. 3.

Writing $(-)^{*}:=[-,\mathcal{D}]_{{\mathsf{Chu}}_{D}(\mathsf{C})}$, the
tensor product of $\mathcal{A}$ and $\mathcal{B}$ is defined as

$\mathcal{A}\otimes_{{\mathsf{Chu}}_{D}(\mathsf{C})}\mathcal{B}:=([\mathcal{A},\mathcal{B}^{*}]_{{\mathsf{Chu}}_{D}(\mathsf{C})})^{*}.$

With this structure in place, we obtain [Bar91, p. 8, Theorem 4.3]:

######  Theorem 3.3.4.3.

Let $\mathsf{C}$ be a bicomplete closed symmetric monoidal category and let
$D$ be an object of $\mathsf{C}$. Then ${\mathsf{Chu}}_{D}(\mathsf{C})$ is a
bicomplete $*$-autonomous category. Moreover, the functor

$\mathsf{C}\hookrightarrow{\mathsf{Chu}}_{D}(\mathsf{C}),\;\;A\mapsto(A,[A,D],e^{A,D}\circ\sigma^{A,[A,D]}),\;f\mapsto(f,[f,D])$

is fully faithful, strongly symmetric monoidal, as well as left adjoint and
right inverse to the functor,

${\mathsf{Chu}}_{D}(\mathsf{C})\to\mathsf{C},\;\;(A,A^{\top},a)\mapsto
A,\;(f,f^{\top})\mapsto f,$

##### The extensional-separated Chu construction

Denoting by $\mathsf{Vect}$ the category of vector spaces over a field $K$,
the objects of ${\mathsf{Chu}}_{K}(\mathsf{Vect})$ are _almost_ what is known
in the context of functional analysis as a _dual system_ (or _paired vector
space_ , see 1.4.4.1). The only difference is that for
$(V,V^{\top},\langle-,-\rangle)$ to be a dual system, one requires the pairing
$\langle-,-\rangle$ to be a _perfect pairing_. This means that both

$V\to(V^{\top})^{\prime},\;\;\;v\mapsto\langle v,-\rangle$

and

$V^{\top}\to V^{\prime},\;\;\;v^{\top}\mapsto\langle-,v^{\top}\rangle$

are to be injective. The injectivity of the first of these maps expresses that
$V^{\top}$ _separates the points_ of $V$. The injectivity of the second map,
on the other hand, enables us to view elements of $V^{\top}$ as genuine
functionals on $V$. Put differently, it expresses _function extensionality_
for elements of $V^{\top}$.

These separation and extensionality requirements can be extended to the
general case of the Chu construction to yield a further way of constructing
$*$-autonomous categories.

######  Definition 3.3.4.4 (Extensional/separated objects).

Let $\mathsf{C}$ be a coregular, closed symmetric monoidal category and let
$D$ be an object of $\mathsf{C}$. An object $(A,A^{\top},a)$ of
${\mathsf{Chu}}_{D}(\mathsf{C})$ is called

  1. 1.

_extensional_ if the morphism,

$\phi:A^{\top}\to[A,D]_{\mathsf{C}},$

induced from $a$ (via the tensor-hom adjunction), is a regular monomorphism,
and

  2. 2.

_separated_ if the morphism,

$\psi:A\to[A^{\top},D]_{\mathsf{C}},$

induced from $a$, is a regular monomorphism.

Define the _separation_ (or _pair separation_ in distinction to the
$D$-separation of 3.3.2.7) of $(A,A^{\top},a)$ as the triple
$(\operatorname*{im}(\psi),A^{\top},\tilde{a})$, where

  1. 1.

$\operatorname*{im}(\psi)$ is the regular image of $\psi$, and

  2. 2.

$\tilde{a}$ is given by the composite,

$\operatorname*{im}(\psi)\otimes_{\mathsf{C}}A^{\top}\to[A^{\top},D]\otimes_{\mathsf{C}}A^{\top}\to
D.$

Similarly, define the _extensionalisation_ as the triple
$(A,\operatorname*{im}(\phi),\overline{a})$, where

  1. 1.

$\operatorname*{im}(\phi)$ is the regular image of $\phi$, and

  2. 2.

$\overline{a}$ is the composite,

$A\otimes_{\mathsf{C}}\operatorname*{im}(\phi)\to
A\otimes_{\mathsf{C}}[A,D]\to D.$

Finally, denote full subcategories of extensional (separated, resp.) objects
of ${\mathsf{Chu}}_{D}(\mathsf{C})$ by $e\mathsf{Chu}_{D}(\mathsf{C})$
($s\mathsf{Chu}_{D}(\mathsf{C})$, resp.), and write

$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$

for the full subcategory of ${\mathsf{Chu}}_{D}(\mathsf{C})$ spanned by those
objects which are both separated _and_ extensional.

######  Remark 3.3.4.5.

In [Bar98], Barr defines the separated-extensional Chu category
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ with respect to arbitrary
factorisation systems. Coregularity of the category $\mathsf{C}$ ensures the
existence of unique regular image factorisations, so that our definition is
that of Barr for the special case of the regular image factorisation system.
Moreover, this factorisation system also has the following two properties that
Barr requires of the given factorisation system $\mathcal{E}/\mathcal{M}$.

  1. 1.

Every morphism in $\mathcal{E}$ is an epimorphism.

  2. 2.

For every object $A$, if $m\in\mathcal{M}$, then
$[A,m]_{\mathsf{C}}\in\mathcal{M}$.

For the regular image factorisation system in a coregular closed symmetric
monoidal category, the first condition holds by definition of this
factorisation system, while the other one follows from then fact that the
internal hom preserves limits in the second variable.

######  Example 3.3.4.6.

The objects of $\widehat{\mathsf{Chu}}_{\mathbb{K}}(\mathsf{Vect})$ are now
exactly dual systems, or “paired vector spaces”, in the sense of functional-
analytic duality theory (mentioned before in the beginning of Section 3.3.4,
see 1.4.4.1 for a definition).

We will need the following facts on the relation between the
extensionalisation and (pair) separation functors $e$ and $s$ [Bar98,
Proposition 3.2, Proposition 3.4]:

######  Proposition 3.3.4.7.

Lt $\mathsf{C}$ be a coregular closed symmetric monoidal category and let $D$
be an object of $\mathsf{C}$. The extensionalisation $e$ and separation $s$
extend to functors that are right (left, resp.) adjoint to the inclusions of
the full subcategories of extensional (separated, resp.) objects.

Moreover, the tensor unit of $\mathsf{Chu}_{D}(\mathsf{C})$ is extensional and
the full subcategory $e\mathsf{Chu}(\mathsf{C})$ of extensional objects is
closed under tensor products. Hence, $e\mathsf{Chu}(\mathsf{C})$ is a
symmetric monoidal category and the inclusion functor from $\mathsf{C}$ into
$\mathsf{Chu}(\mathsf{C})$ co-restricts to a strong symmetric monoidal functor

$\mathsf{C}\to e\mathsf{Chu}(\mathsf{C}).$

We now describe the closed monoidal structure on
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$.

######  Definition 3.3.4.8.

Define

  1. 1.

the dualising object and tensor unit of
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ as $e\mathcal{D}$ and $s\mathcal{I}$,
respectively (where $\mathcal{D}$ and $\mathcal{I}$ are those of
${\mathsf{Chu}}_{D}(\mathsf{C})$),

  2. 2.

the internal hom of $\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ as

$[\mathcal{A},\mathcal{B}]_{\widehat{\mathsf{Chu}}_{D}(\mathsf{C})}:=e[\mathcal{A},\mathcal{B}]_{{\mathsf{Chu}}_{D}(\mathsf{C})},$

  3. 3.

and the tensor product as

$\mathcal{A}\otimes_{\widehat{\mathsf{Chu}}_{D}(\mathsf{C})}\mathcal{B}:=s(\mathcal{A}\otimes_{{\mathsf{Chu}}_{D}(\mathsf{C})}\mathcal{B}).$

######  Theorem 3.3.4.9.

Let $\mathsf{C}$ be a coregular bicomplete closed symmetric monoidal category
and let $D$ be an object. Then, with the structure described in 3.3.4.8,
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ is a bicomplete $*$-autonomous
category. Moreover, the functor

$\mathsf{C}\to\widehat{\mathsf{Chu}}_{D}(\mathsf{C}),\;\;A\mapsto
s(A,[A,D],e^{A,D}\circ\sigma^{A,[A,D]})$

is a strong symmetric monoidal functor.

###### Proof.

For $*$-autonomy of $\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$, see [Bar98]. For
the second part of the statement, notice the above functor is the composite of
the functors

${{\mathsf{C}}}$${{e\mathsf{Chu}(\mathsf{C})}}$${{\widehat{\mathsf{Chu}}_{D}(\mathsf{C}),}}$$\scriptstyle{s}$

the first of which is strong symmetric monoidal by 3.3.4.7. The second, on the
other hand, is a _strict_ monoidal functor, by definition of the tensor
product on $\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$:

$\mathcal{A}\otimes_{\widehat{\mathsf{Chu}}_{D}(\mathsf{C})}\mathcal{B}=s(\mathcal{A}\otimes_{{\mathsf{Chu}}_{D}(\mathsf{C})}\mathcal{B})=s(\mathcal{A}\otimes_{e\mathsf{Chu}_{D}(\mathsf{C})}\mathcal{B})$

The claim now follows from the fact that a composite of strong symmetric
monoidal functors is strong symmetric monoidal. ∎

One important difference between the categories
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ and ${\mathsf{Chu}}_{D}(\mathsf{C})$
is the following.

######  Proposition 3.3.4.10.

The functor

$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})\to\mathsf{C},\;\;\;(A,A^{\top},a)\mapsto
A$

is faithful.

###### Proof.

If $(f,f_{0}^{\top}),(f,f_{1}^{\top}):(A,A^{\top},a)\to(B,B^{\top},b)$ are two
morphisms in $\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ inducing the same
morphism $f$ in $\mathsf{C}$ under the above functor, then, since the diagram

${{B^{\top}}}$${{A^{\top}}}$${{[B,D]}}$${{[B,D]}}$$\scriptstyle{f_{0}^{\top}}$$\scriptstyle{[f,D]}$$\scriptstyle{f_{1}^{\top}}$

commutes and the morphism $A^{\top}\to[A,D]$ is a monomorphism,
$f_{0}^{\top}=f_{1}^{\top}$. ∎

We may hence interpret the above functor as a forgetful functor. Under this
interpretation, the objects of $\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ are
$\mathsf{C}$-objects with some additional structure (which the forgetful
functor “forgets”), and the morphisms in
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ are precisely those
$\mathsf{C}$-morphisms between (underlying $\mathsf{C}$-objects of) objects of
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ which preserve this additional
structure.

In particular, if $\mathsf{C}$ is a concrete category (i.e. equipped with a
faithful functor to the category of sets), then so is
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ in a canonical way. This is in sharp
contrast to the larger category ${\mathsf{Chu}}_{D}(\mathsf{C})$. For example,
the triple $(0,V,0)$, where $V$ is _any_ (algebraic) vector space, is an
object of ${\mathsf{Chu}}_{D}(\mathsf{Vect}(\mathsf{Set}))$. Such “degenerate”
examples cannot occur in
$\widehat{\mathsf{Chu}}_{D}(\mathsf{Vect}(\mathsf{Set}))$.

##### Relation between the separated-extensional Chu construction and
$D$-separated objects

A natural question is now which objects of $\mathsf{C}$ can arise as the
underlying $\mathsf{C}$-object of an an object of
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$. In other words: which
$\mathsf{C}$-objects admit the structure of an object of
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$? These are precisely the
$D$-separated objects.

######  Proposition 3.3.4.11.

Let $\mathsf{C}$ be a coregular bicomplete closed symmetric monoidal category
and let $D\in\mathsf{C}$. Then an object $A$ of $\mathsf{C}$ is $D$-separated
if, and only if, it is the underlying $\mathsf{C}$-object of an object of
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ (i.e. there is an object of
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ of the form $(A,A^{\top},a)$). As a
consequence, the functor

$\iota:\mathsf{sep}_{D}(\mathsf{C})\hookrightarrow\widehat{\mathsf{Chu}}_{D}(\mathsf{C}),\;\;A\mapsto(A,[A,D],e^{A,D}\circ\sigma^{A,[A,D]}),\;f\mapsto(f,[f,D])$

is well-defined, fully faithful and left adjoint as well as right inverse to

$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})\to\mathsf{sep}_{D}(\mathsf{C}),\;\;\;(A,A^{\top},a)\mapsto
A.$

In other words, $\mathsf{sep}_{D}(\mathsf{C})$ is a coreflective subcategory
of $\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$ and we have the following diagram
of functors, in which the solid square of inclusions commutes:

${{\mathsf{sep}_{D}(\mathsf{C})}}$${{\widehat{\mathsf{Chu}}_{D}(\mathsf{C})}}$${{\mathsf{C}}}$${{\mathsf{Chu}_{D}(\mathsf{C})}}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$

###### Proof.

If $A\in\mathsf{C}$ is $D$-separated, then
$(A,[A,D],e^{A,D}\circ\sigma^{A,[A,D]})$ is an object of
$\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$. Conversely, if
$(A,A^{\top},a)\in\widehat{\mathsf{Chu}}_{D}(\mathsf{C})$, then we have a
regular monomorphism,

$A\to[A^{\top},D],$

and by 3.3.2.6, $A$ is $D$-separated. The other claims now follow from
3.3.4.3. ∎

####  3.3.5. Closed symmetric monoidal categories over cartesian closed
categories

The following notion captures when a given closed symmetric monoidal (or
$*$-autonomous, in particular) category stands in a good relationship with a
cartesian closed “base category”.

######  Definition 3.3.5.1.

A _closed-symmetric-monoidal-over-cartesian-closed category_ $|\cdot|\vdash
T:\mathsf{D}\to\mathsf{C}$ is a closed symmetric monoidal category
$\mathsf{D}$, together with a cartesian closed category $\mathsf{C}$ (the
_base category_) and a symmetric monoidal adjunction (3.2.2.3),

${{\mathsf{C}}}$${{\mathsf{D}}}$$\scriptstyle{U}$$\scriptstyle{T}$$\scriptstyle{\dashv}$

By a bicomplete, coregular closed-symmetric-monoidal-over-cartesian-closed
category $|\cdot|\vdash T:\mathsf{D}\to\mathsf{C}$ we mean a closed-symmetric-
monoidal-over-cartesian-closed category for which both $\mathsf{D}$ and
$\mathsf{C}$ are bicomplete (i.e. have all limits and colimits) and coregular.
A _$*$ -autonomous-over-cartesian-closed category_ is a closed-symmetric-
monoidal-over-cartesian-closed category $|\cdot|\vdash
T:\mathsf{D}\to\mathsf{C}$ where $\mathsf{D}$ is $*$-autonomous.

We cal call $\mathsf{C}$ the base category and the adjunction $|\cdot|\vdash
T$ the free-forgetful adjunction.

######  Remark 3.3.5.2.

A definition similar to 3.3.5.1 is given in [dP14, p. 12, Definition 6] under
the name _linear-non-linear category_ , in the context of (the categorical
semantics of) _linear logic_.

In our context, one should think of $\mathsf{D}$ as a category of linear
spaces (e.g. vector spaces or linear $hk$-spaces) living over a base category
$\mathsf{C}$ of “general” spaces (e.g. sets or $hk$-spaces).

The free-forgetful adjunction defining a closed-symmetric-monoidal-over-
cartesian-closed category is always enriched over the base category:

######  Proposition 3.3.5.3.

Let $|\cdot|\vdash T:\mathsf{D}\to\mathsf{C}$ be a closed-symmetric-monoidal-
over-cartesian-closed category. Then we have a natural isomorphism,

$[X,V]_{\mathsf{C}}\cong|[TX,V]_{\mathsf{D}}|.\qquad(X\in\mathsf{C},V\in\mathsf{D})$

###### Proof.

Using that the free-forgetful adjunction is a monoidal adjunction, we obtain a
sequence of natural isomorphisms,

$\displaystyle\operatorname{Hom}_{\mathsf{C}}(Y,[X,V]_{\mathsf{C}})$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(Y\times X,V)$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{D}}(T(Y\times X),V)$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{D}}(TY\otimes_{\mathsf{D}}TX,V)$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{D}}(TY,[TX,V]_{\mathsf{D}})$
$\displaystyle\cong\operatorname{Hom}_{\mathsf{C}}(Y,|[TX,V]_{\mathsf{D}}|).\qquad(Y\in\mathsf{C})$

The claim now follows from the Yoneda lemma. ∎

##### Free $D$-separated objects

The notion of $D$-separated object is compatible with that of closed-
symmetric-monoidal-over-cartesian-closed category, in the following sense.

######  Proposition 3.3.5.4.

Let $|\cdot|\vdash T:\mathsf{D}\to\mathsf{C}$ be a closed-symmetric-monoidal-
over-cartesian-closed category and let $D\in\mathsf{D}$ be an object. Then

$T_{D}:=\mathsf{sep}_{D}\circ T$

is left adjoint to the “forgetful” functor

$|\cdot|\circ\iota:\mathsf{sep}_{D}\mathsf{D}\to\mathsf{C},$

where

$\iota:\mathsf{sep}_{D}\mathsf{D}\to\mathsf{D}$

is the inclusion functor, and

$|\cdot|\circ\iota\vdash T_{D}:\mathsf{D}\to\mathsf{C}$

is again a closed-symmetric-monoidal-over-cartesian-closed category.

###### Proof.

The $D$-separation $\mathsf{sep}_{D}$ is left adjoint to the inclusion functor
$\iota$ and this is a symmetric monoidal adjunction, since $D$-separated
objects form an exponential ideal in $\mathsf{D}$. Composing the two symmetric
monoidal adjunctions $\iota\vdash\mathsf{sep}_{D}$ and $|\cdot|\vdash T$ gives
the desired

$|\cdot|\circ\iota\vdash T_{D}.$

∎

##### The free replete linear $hk$-space

As a particular case, the category $r{\mathsf{Vect}(hk\mathsf{Top})}$ of
replete linear $hk$-spaces also admits _free objects_.

######  Definition 3.3.5.5 (Free replete linear $hk$-space).

Let $X$ be an $hk$-space. Define the _free linear $hk$-space_ on $X$,

$\mathcal{M}_{c}(X):=rF_{\mathbb{K}}(X),$

as the repletion of the free linear $hk$-space on $X$.

######  Remark 3.3.5.6.

The notation $\mathcal{M}_{c}(X)$ will be justified in Chapter 4, where we
will show that this object can be identified with the space of compactly
supported Radon measures on $X$, when $X$ is a Hausdorff $k$-space.

######  Warning 3.3.5.7.

We will use the same notation for the free _paired_ linear $hk$-space (and
have already done so in the introduction). This unproblematic, since these two
objects agree, in the sense that the free replete linear $hk$-space
$\mathcal{M}_{c}(X)$ carries a natural paired linear $hk$-space structure
under which it becomes the free paired linear $hk$-space.

######  Corollary 3.3.5.8 (Universal property of $\mathcal{M}_{c}(X)$).

Let $X$ be an $hk$-space, let $V$ be a replete linear $hk$-space and $f:X\to
V$ be a continuous map. Then there exists a unique continuous linear map
$\tilde{f}:\mathcal{M}_{c}(X)\to V$ making the diagram

${{\mathcal{M}_{c}(X)}}$${V}$${X}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\delta_{\bullet}}$$\scriptstyle{f}$

commute, where $\delta_{\bullet}$ is the composite of the canonical maps,

$X\to F_{\mathbb{K}}(X)\to rF_{\mathbb{K}}(X).$

##### A construction theorem for $*$-Autonomous-over-cartesian-closed
categories

The following is a general construction theorem which uses the separated-
extensional Chu construction to obtain a $*$-autonomous-over-cartesian-closed
category $U\vdash M_{D}:\widehat{\mathsf{Chu}}_{D}(\mathsf{D})\to\mathsf{C}$
from a general closed-symmetric-monoidal-over-cartesian-closed category
$|\cdot|\vdash T:\mathsf{D}\to\mathsf{C}$ in such a way that if
$|\cdot|:\mathsf{D}\to\mathsf{C}$ is a faithful functor, then so is
$U:\widehat{\mathsf{Chu}}_{D}(\mathsf{D})\to\mathsf{C}$.

######  Theorem 3.3.5.9.

Let $|\cdot|\vdash T:\mathsf{D}\to\mathsf{C}$ be a bicomplete, coregular
closed-symmetric-monoidal-over-cartesian-closed category $\mathsf{C}$, and let
$D\in\mathsf{D}$ be an object. Then the forgetful functor,

$U:\widehat{\mathsf{Chu}}_{D}(\mathsf{D})\to\mathsf{C},\;\;\;(A,A^{\top},\langle-,-\rangle)\mapsto|A|,$

has a strong symmetric monoidal left adjoint $M_{D}$. Hence, the resulting
adjunction,

${{\widehat{\mathsf{Chu}}_{D}(\mathsf{D})}}$${{\mathsf{C}},}$$\scriptstyle{U}$$\scriptstyle{M_{D}}$

defines a $*$-autonomous-over-cartesian-closed category.

###### Proof.

By 3.3.2.10 and 3.3.4.11, we have the following adjunctions,

${{\mathsf{C}}}$${{\mathsf{D}}}$${{\mathsf{sep}_{D}(\mathsf{D})}}$${{\widehat{\mathsf{Chu}}_{D}(\mathsf{D})}}$$\scriptstyle{T}$$\scriptstyle{\mathsf{sep}_{D}}$$\scriptstyle{\iota}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$$\scriptstyle{\dashv}$

Hence,

$M_{D}:=\iota\circ\mathsf{sep}_{D}\circ T$

is the desired left adjoint. It remains to show that $M_{D}$ is (or rather,
admits the structure of a) strong symmetric monoidal functor. To see this,
notice that $M_{D}$ also factors as the composite of two strong symmetric
monoidal functors,

${{\mathsf{C}}}$${{\mathsf{D}}}$${{\widehat{\mathsf{Chu}}_{D}(\mathsf{D}),}}$$\scriptstyle{T}$

where the functor on the right is the functor as described in 3.3.4.9. ∎

##### Paired linear $hk$-spaces

Applying the extensional-separated Chu construction to the category of linear
$hk$-spaces, we obtain _paired linear $hk$-spaces_.

######  Definition 3.3.5.10.

A _paired linear $hk$-space_ is an object of
$\widehat{\mathsf{Chu}}_{\mathbb{K}}({\mathsf{Vect}(hk\mathsf{Top})})$, i.e. a
linear $hk$-space $V$ together with a further linear $hk$-space $V^{*}$ and a
bilinear map

$\langle-,-\rangle:V\times V^{*}\to\mathbb{K}$

such that both

$V\to(V^{*})^{\wedge},\;\;x\mapsto\langle x,-\rangle$

and

$V^{*}\to V^{\wedge},\;\;\phi\mapsto\langle-,\phi\rangle$

are regular monomorphisms in ${\mathsf{Vect}(hk\mathsf{Top})}$.

Since, by 3.3.1.5, regular monomorphisms in ${\mathsf{Vect}(hk\mathsf{Top})}$
are equivalently closed embeddings, we can reformulate this definition as
follows, identifying $V^{*}$ with its image in $V^{\wedge}$.

######  Definition 3.3.5.11 (Paired linear $hk$-space, more explicitly).

A _paired linear $hk$-space_ is a linear $hk$-space $V$ together with a closed
linear subspace $V^{*}\subseteq V^{\wedge}$ of the natural dual such that

$V\to(V^{*})^{\wedge},\;\;x\mapsto\langle x,-\rangle$

is a closed embedding. A morphism of paired linear $hk$-spaces $V,W$ is a
continuous linear map $f:V\to W$ such that $f^{\wedge}(W^{*})\subseteq V^{*}$,
where

$f^{\wedge}:W^{\wedge}\to V^{\wedge},\;\;\phi\mapsto\phi\circ f$

is the adjoint of $f$. We denote the category of paired linear $hk$-spaces by

$\mathsf{L}:=\widehat{\mathsf{Chu}}_{\mathbb{K}}({\mathsf{Vect}(hk\mathsf{Top})}).$

We denote the internal hom and tensor product of paired linear $hk$-spaces
simply by

$[V,W]:=[V,W]_{\mathsf{L}},\;\;V\,\widehat{\otimes}\,W:=V\otimes_{\mathsf{L}}W\qquad\qquad(V,W\in\mathsf{L})$

as long as there is no potential for ambiguity. The underlying linear
$hk$-space of $[V,\mathbb{K}]$ is exactly $V^{*}$, so it is justified to also
write

$V^{*}:=[V,\mathbb{K}]\qquad\qquad(V\in\mathsf{L})$

for the dual, viewed as a paired linear $hk$-space.

As a special case of 3.3.5.9, we immediately obtain that $\mathsf{L}$ is a
$*$-autonomous category with respect to the above tensor product and internal
hom.

######  Remark 3.3.5.12.

By construction, the internal hom $[V,W]$ of paired linear $hk$-spaces $V,W$
is the closed subspace

$[V,W]\subseteq L(V,W)$

of the space of continuous linear maps $L(V,W)$ (the internal hom in
${\mathsf{Vect}(hk\mathsf{Top})}$ and also $r{\mathsf{Vect}(hk\mathsf{Top})}$)
consisting of those continuous linear maps which are morphisms of paired
linear $hk$-spaces.

##### The category of replete linear $hk$-spaces embeds into the category of
paired linear $hk$-spaces

A wide range of examples of paired linear $hk$-spaces is supplied by
considering replete linear $hk$-spaces as paired linear $hk$-spaces, as
follows.

######  Proposition 3.3.5.13.

The functor

$\iota:r{\mathsf{Vect}(hk\mathsf{Top})}\hookrightarrow\mathsf{L},\;\;V\mapsto(V,V^{\wedge}),\;f\mapsto
f^{\wedge}$

is fully faithful, displaying $r{\mathsf{Vect}(hk\mathsf{Top})}$ as a
coreflective subcategory of $\mathsf{L}$.

###### Proof.

This is a special case of 3.3.4.11. ∎

As a consequence, when $W$ is any paired linear $hk$-space and $V$ is in the
essential image of the inclusion functor,

$r{\mathsf{Vect}(hk\mathsf{Top})}\hookrightarrow\mathsf{L},$

i.e. $V^{*}=V^{\wedge}$, then every continuous linear map $V\to W$ is a
morphism of paired linear $hk$-spaces and we have,

$[V,W]=L(V,W).$

##### Fréchet and Brauner spaces are paired linear $k$-spaces in a unique way

While for a general replete linear $hk$-space $V$, there may a priori be
several paired linear $hk$-spaces whose underlying linear $hk$-space is $V$,
this is not possible for Fréchet or Brauner spaces: their dual is, in some
sense, uniquely determined.

######  Proposition 3.3.5.14.

Let $V$ be a paired linear $hk$-space with dual $V^{*}$ and suppose that the
underlying linear $hk$-space of $V$ is a Fréchet space or a Brauner space.
Then $V^{*}=V^{\wedge}$.

###### Proof.

Notice that the assumption that

$j:V\to(V^{*})^{\wedge},\;\;\;x\mapsto\langle x,-\rangle$

is a closed embedding implies that

$j^{\wedge}:(V^{*})^{\wedge\wedge}\to
V^{\wedge},\;\;\;\chi\mapsto(x\mapsto\chi(\langle x,-\rangle))$

is surjective, using the Hahn-Banach extension theorem (which is applicable
because both $V$ and $(V^{*})^{\wedge}$ are Fréchet/Brauner spaces). By Smith
duality, the canonical map

$\eta^{V^{*}}:V^{*}\to(V^{*})^{\wedge\wedge}$

is an isomorphism. Hence, observing that

$j^{\wedge}\circ\eta^{V^{*}}:V^{*}\to V^{\wedge},\;\;\phi\mapsto\phi,$

is the embedding of $V^{*}$ into $V^{\wedge}$, we see that this map is
surjective and hence the identity. ∎

##### The free paired linear $hk$-space

By 3.3.5.9, the forgetful functor $\mathsf{L}\to hk\mathsf{Top}$ admits a left
adjoint, and this left adjoint is given by the composition of the left adjoint
$\mathcal{M}_{c}(X)$ to the forgetful functor
$r{\mathsf{Vect}(hk\mathsf{Top})}\to hk\mathsf{Top}$, followed by the
inclusion of $r{\mathsf{Vect}(hk\mathsf{Top})}$ into $\mathsf{L}$. Hence, we
are justified in defining (see also 3.3.5.7):

######  Definition 3.3.5.15.

We denote the left adjoint to the forgetful functor $\mathsf{L}\to
hk\mathsf{Top}$ by $\mathcal{M}_{c}$. For an $hk$-space $X$, we call
$\mathcal{M}_{c}(X)$ the _free paired linear $hk$-space_ on $X$.

######  Remark 3.3.5.16.

The notation $\mathcal{M}_{c}(X)$ will be justified in Chapter 4, where we
will show that the elements of $\mathcal{M}_{c}(X)$ can be identified with
certain compactly supported measures. Moreover, by 3.3.5.3, the free-forgetful
adjunction between paired linear $hk$-spaces and $hk$-spaces is
$hk\mathsf{Top}$-enriched,

$[\mathcal{M}_{c}(X),V]\cong C(X,V),\qquad(X\in
hk\mathsf{Top},V\in\mathsf{L})$

and we will see in Chapter 4 that this natural isomorphism is given by vector-
valued integration (see Remark 4.6.1.4).

######  Remark 3.3.5.17.

Since by 3.3.5.9, the free-forgetful adjunction between paired linear
$hk$-spaces and $hk$-spaces is a symmetric monoidal adjunction, it induces a
symmetric monoidal, i.e. commutative, monad on $hk\mathsf{Top}$ with
underlying endofunctor $X\mapsto\mathcal{M}_{c}(X)$.

##  Chapter 4 Monadic Measure and Integration Theory

In this chapter, we will harvest the fruits of the general abstract
developments of Chapter 3. To this end, we will first define the linear
$hk$-space $C_{b}(X)$ of continuous bounded functions on an $hk$-space $X$.
Then, we will construct $\mathcal{M}(X)$ as a space of continuous linear
functionals on $C_{b}(X)$ and endow $\mathcal{M}(X)$ as well as $C_{b}(X)$
with the structure of a paired linear $hk$-space, rendering them mutually
dual. In Section 4.2, a version of the Riesz representation theorem will allow
us to identify the elements of $\mathcal{M}(X)$ with a certain class of
measures which we call _$k$ -regular measures_. Next, we will construct the
monad structure of $\mathcal{M}$ and the probability monad $\mathcal{P}$ in
Sections 4.3 and 4.4. Finally, we will show in Section 4.6 how the formalism
of (paired or replete) linear $hk$-spaces can be used to obtain a very
coherent theory of vector-valued integration, _monadic vector-valued
integration_.

###  4.1. The Paired Linear $hk$-space $C_{b}(X)$ and its Dual
$\mathcal{M}(X)$

####  4.1.1. The linear $hk$-space $C_{b}(X)$

First, we need to define $C_{b}(X)$ as a linear $hk$-space. (The _paired_
linear structure on $C_{b}(X)$ will come later.)

######  Definition 4.1.1.1.

We define the linear $hk$-space $C_{b}(X)$ of bounded continuous
($\mathbb{K}$-valued) functions on an $hk$-space $X$ as the filtered colimit
(in the category of $hk$-spaces),

$C_{b}(X):=\operatorname*{colim}_{n\in\mathbb{N}}\>(nD)^{X},$

of the diagram,

$D^{X}\hookrightarrow(2D)^{X}\hookrightarrow(3D)^{X}\hookrightarrow...,$

where

$D=\\{\lambda\in\mathbb{K}\mid|\lambda|\leq 1\\}\subseteq\mathbb{K}$

is the unit disk.

######  Remark 4.1.1.2.

Since the underlying set of this colimit can be identified with the union of
all functions whose absolute value is bounded by some natural number, the
elements of $C_{b}(X)$ correspond exactly to bounded continuous functions on
$X$, justifying the notation. Moreover, (pointwise) addition and scalar
multiplication are continuous with respect to the above colimit topology.
Hence, $C_{b}(X)$ is indeed a linear $hk$-space.

Convergent sequences in $C_{b}(X)$ are simple to describe.

######  Lemma 4.1.1.3.

Let $X$ be an $hk$-space. Then a sequence $(f_{n})$ of continuous bounded
functions converges to $f$ in $C_{b}(X)$ if, and only if, $(f_{n})$ is
uniformly bounded and converges in the compact-open topology to $f$.

###### Proof.

A convergent sequence in $C_{b}(X)$ can be identified with a continuous map

$\mathbb{N}\cup\\{\infty\\}\to C_{b}(X)$

from the one-point compactification of the natural numbers to $C_{b}(X)$. The
image of such map being compact, 2.1.4.13 implies that every convergent
sequence lies in one of the spaces $(nD)^{X}$ (for some $n\in\mathbb{N}$) and
is hence uniformly bounded. Moreover, a sequence converges in $(nD)^{X}$ if,
and only if, it converges in the compact-open topology (see 2.1.5.1). ∎

We will need the following lemma on the continuous linear functionals on
$C_{b}(X)$ later.

######  Lemma 4.1.1.4.

Let $X$ be an $hk$-space and let $\phi$ be a (not necessarily continuous)
functional on $C_{b}(X)$. Suppose that for every uniformly bounded net of
continuous functions $(f_{i})$ on $X$ converging in the compact-open topology
to $0$, $\phi(f_{i})\to 0$. Then $\phi$ is continuous.

###### Proof.

By assumption, $\phi$ is continuous on each space of continuous functions
$C_{c.o.}(X,nD)$ with the compact-open topology, implying that it is also
continuous on each $(nD)^{X}$. The claim follows, since $C_{b}(X)$ is given as
the colimit over these spaces. ∎

####  4.1.2. The linear $hk$-space $\mathcal{M}(X)$

At this point, as for $C_{b}(X)$, we introduce $\mathcal{M}(X)$ only as a
linear $hk$-space. Later, $\mathcal{M}(X)$ will also be given a paired linear
structure, rendering it dual to $C_{b}(X)$. Moreover, while the following
definition of $\mathcal{M}(X)$ is in terms of functionals, we will later
identify $\mathcal{M}(X)$ with a space of measures.

######  Definition 4.1.2.1.

Define

$\mathcal{M}(X):=\overline{\text{span}\\{\delta_{x}\in
C_{b}(X)^{\wedge}\mid\,x\in X\\}}\subseteq C_{b}(X)^{\wedge}$

as the closed linear subspace of the natural dual $C_{b}(X)^{\wedge}$ of
$C_{b}(X)$ generated by the point evaluations (“Dirac functionals”,
$\delta_{x}(f):=f(x)$).

This definition ensures from the start that finitely supported measures will
be dense in $\mathcal{M}(X)$ (once we have shown how to identify elements of
$\mathcal{M}(X)$ with measures), a very convenient feature. (For instance, we
will use this in the proof of 4.3.2.2, the commutativity of $\mathcal{M}$ as a
monad.) Moreover, as a closed linear subspace of a replete linear $hk$-space,
$\mathcal{M}(X)$ is a replete linear $hk$-space.

####  4.1.3. The natural dual of $\mathcal{M}(X)$ is $C_{b}(X)$

The next lemma will be needed for showing that $\mathcal{M}(X)^{\wedge}\cong
C_{b}(X)$, which we will then use to define the paired linear structure on
both spaces in an obvious way.

######  Lemma 4.1.3.1.

Let $X$ be an $hk$-space. Then the map

$\delta_{\bullet}:X\to\mathcal{M}(X),\;\;x\mapsto\delta_{x}$

is continuous.

###### Proof.

Since $\mathcal{M}(X)$ was defined as a subspace of $C_{b}(X)^{\wedge}$, it
suffices to show that the map

$X\to C_{b}(X)^{\wedge},\;\;x\mapsto\delta_{\bullet}$

is continuous. This map factors as the composite,

${X}$${{C(X)^{\wedge}}}$${{C_{b}(X)^{\wedge}}}$$\scriptstyle{\delta_{\bullet}}$$\scriptstyle{(-)|_{C_{b}(X)}}$

so its continuity follows from cartesian closure of $hk\mathsf{Top}$ and the
fact that the inclusion map $C_{b}(X)\to C(X)$ is continuous. ∎

######  Proposition 4.1.3.2.

Let $X$ be an $hk$-space. Then the linear maps,

$\Phi:\mathcal{M}(X)^{\wedge}\to C_{b}(X),\;\;\;F\mapsto(x\mapsto
F(\delta_{x})),$

and,

$\Psi:C_{b}(X)\to\mathcal{M}(X)^{\wedge},\;\;\;f\mapsto(\mu\mapsto\mu(f)),$

are continuous and mutually inverse and hence constitute isomorphisms (natural
in $X$):

$\mathcal{M}(X)^{\wedge}\cong C_{b}(X).$

###### Proof.

Continuity of $\Psi$ is clear from cartesian closure of $hk\mathsf{Top}$, as
is continuity of $\Phi$ when taking 4.1.3.1 into account. Now, for any $f\in
C_{b}(X)$, $x\in X$,

$\displaystyle\Phi\circ\Psi(f)(x)=\Phi(\mu\mapsto\mu(f))(x)$
$\displaystyle=\delta_{x}(f)=f(x)=\mathsf{id}_{C_{b}(X)}(f)(x).$

Similarly, for all $F\in\mathcal{M}(X)^{\wedge}$, $x_{0}\in X$,

$\displaystyle\Psi\circ\Phi(F)(\delta_{x_{0}})$ $\displaystyle=\Psi(x\mapsto
F(\delta_{x}))(\delta_{x_{0}})$ $\displaystyle=\delta_{x_{0}}(x\mapsto
F(\delta_{x}))$
$\displaystyle=F(\delta_{x_{0}})=\mathsf{id}_{\mathcal{M}(X)^{\wedge}}(F)(\delta_{x_{0}}),$

which, since the span of all such $\delta_{x_{0}}$ is dense in
$\mathcal{M}(X)$, implies the claim. ∎

####  4.1.4. $C_{b}(X)$ and $\mathcal{M}(X)$ as paired linear $hk$-spaces

Finally, we may now endow $C_{b}(X)$ and $\mathcal{M}(X)$ with (mutually dual)
paired linear structures.

######  Definition 4.1.4.1.

We define the paired linear $hk$-space $\mathcal{M}(X)$ as equipped with its
natural dual,

$\mathcal{M}(X)^{*}:=\mathcal{M}(X)^{\wedge}.$

Accordingly, we define the paired linear $hk$-space $C_{b}(X)$ as being
equipped with the admissible dual

$C_{b}(X)^{*}:=\mathcal{M}(X).$

We hence obtain two functors,

$C_{b}(X):hk\mathsf{Top}^{\text{op}}\to\mathsf{L},\;\;\;X\mapsto
C_{b}(X),\;f\mapsto(-\circ f),$
$\mathcal{M}(X):hk\mathsf{Top}\to\mathsf{L},\;\;\;X\mapsto\mathcal{M}(X),\;f\mapsto
f_{*}.$

###  4.2. $k$-Regular Measures

####  4.2.1. Some measure-theoretic preliminaries

Up to this point, we only know that the spaces $\mathcal{M}(X)$ and
$\mathcal{M}_{c}(X)$ are spaces of _functionals_ , but we really want them to
be spaces of genuine _measures_. To finally bring measure theory into the
picture, we remind ourselves of some notions from the theory of measures on
topological spaces. We follow the terminology of [BR07].

######  Warning 4.2.1.1.

By default, “measure” will mean “countably additive $\mathbb{K}$-valued
measure of bounded variation”.

######  Definition 4.2.1.2.

Let $X$ be a topological space. The _Baire $\sigma$-algebra_ is the
$\sigma$-algebra generated by all continuous $\mathbb{K}$-valued functions on
$X$ (it does not matter whether $\mathbb{K}=\mathbb{R}$ or
$\mathbb{K}=\mathbb{C}$). A _Baire measure_ is a measure on the Baire
$\sigma$-algebra. Similarly, recall that a _Borel measure_ is a measure
defined on the _Borel $\sigma$-algebra_, which, by definition, is generated by
the open subsets of $X$. A Borel measure $\mu$ on $X$ is a _Radon measure_ if
for every $\epsilon>0$, there is a compact subset $K_{\epsilon}\subseteq X$
such that $|\mu|(X\mathbin{\backslash}K_{\epsilon})<\epsilon$.

When $X$ is sufficiently well-behaved, the notions of Baire and Radon measure
essentially coincide:

######  Theorem 4.2.1.3.

On a Polish space, the Baire and Borel $\sigma$-algebras agree and every Baire
(equivalently Borel) measure is a Radon measure. Moreover, on a
$\sigma$-compact completely regular space, every Baire measure has a unique
extension to a Radon measure.

###### Proof.

For the statement about Polish spaces, see [BR07, p. 70, Theorem 7.1.7], for
the one on compacta, see [BR07, p. 81, Theorem 7.3.4]. ∎

Hence, if a space $X$ is either a Polish or compact, Baire measures on $X$ can
be identified with Radon measures.

####  4.2.2. A Riesz-Markov-type theorem

The proposition that follows is a first step towards our desired
representation theorem for $\mathcal{M}(X)$ by representing functionals from
the larger space $C_{b}(X)^{\wedge}$ as Baire measures.

######  Proposition 4.2.2.1.

Let $X$ be an $hk$-space. Then for every $\phi\in C_{b}(X)^{\wedge}$, there
exists a unique Baire measure $\mu$ such that

$\phi(f)=\int f\mathop{}\\!\mathrm{d}\mu,$

for all $f\in C_{b}(X)$.

###### Proof.

Let $\phi\in C_{b}(X)^{\wedge}$. By [BR07, p. 111, Theorem 7.10.1], the
conclusion of the lemma follows if, and only, if for every monotonically
decreasing sequence $(f_{n})$ in $C_{b}(X)$ converging pointwise to zero,
$\phi(f_{n})\to 0$, as well. So let $(f_{n})$ be such a sequence. By Dini’s
theorem, $(f_{n})$ converges to $0$ uniformly on compact subsets of $X$.
Moreover, since it is monotonically decreasing, $(f_{n})$ is uniformly bounded
by the constant $\sup_{x\in X}f_{0}(x)$. Hence, by 4.1.1.3, $(f_{n})$
converges to $0$ in $C_{b}(X)$ and by continuity of $\phi$, $\phi(f_{n})\to
0$, which is what we wanted to show. ∎

####  4.2.3. $k$-Pre-regular measures

The class of measures which represent functionals from $\mathcal{M}(X)$ will
be referred to as _$k$ -regular measures_ (this will be 4.2.4.1). As a
stepping stone towards this definition and towards understanding this class of
measures, we first introduce _$k$ -pre-regular measures_.

######  Definition 4.2.3.1.

Let $X$ be an $hk$-space. A _$k$ -pre-regular measure_ is a Baire measure
$\mu$ on $X$ such that the map

$C_{b}(X)\to\mathbb{K},\;\;f\mapsto\int f\mathrm{d}\mu$

is continuous. Denote the set of $k$-pre-regular measures on $X$ by
$\mathcal{M}_{\mathrm{pre}}(X)$.

As an immediate corollary to 4.2.2.1, we obtain:

######  Corollary 4.2.3.2.

Let $X$ be an $hk$-space. Then the mapping,

$\mathcal{M}_{\mathrm{pre}}(X)\to
C_{b}(X)^{\wedge},\;\;\;\mu\mapsto\int(-)\mathop{}\\!\mathrm{d}\mu,$

is a bijection, so we may identify functionals in $C_{b}(X)^{\wedge}$ with
$k$-pre-regular measures under this bijection.

######  Definition 4.2.3.3.

Let $X$ be an $hk$-space. Denote the closed subspace of
$\mathcal{M}_{\mathrm{pre}}(X)=C(X)^{\wedge}$ consisting of $k$-pre-regular
_probability_ measures (equivalently, positive normalised functionals) by
$\mathcal{P}_{\mathrm{pre}}(X)\subseteq\mathcal{M}_{\mathrm{pre}}(X)$.

In many cases, _any_ Baire measure will be $k$-pre-regular:

######  Proposition 4.2.3.4.

Let $X$ be an $hk$-space and let $\mu$ be a Baire measure $X$ that is the
restriction of some Radon measure $\tilde{\mu}$ on $X$ to the Baire
$\sigma$-algebra. Then $\mu$ is a $k$-pre-regular measure. In particular, if
$X$ either a completely regular $\sigma$-compact $hk$-space (e.g. a compact
Hausdorff space, or a Brauner space), or a Polish space, then _every_ Baire
measure is a $k$-pre-regular measure.

###### Proof.

Let $(f_{i})$ be a net of continuous functions on $X$ uniformly bounded by
$C>0$ converging uniformly on compact subsets to $0$. By 4.1.1.4, it suffices
to show that

$\int f_{i}\mathop{}\\!\mathrm{d}\mu\to 0.$

Let $\epsilon>0$. Then, since $\tilde{\mu}$ is a Radon measure, there exists a
compact subset $K$ such that $|\tilde{\mu}|(X\mathbin{\backslash}K)<\epsilon$.
Now, by the compact convergence of $(f_{i})$, there is an index $i_{0}$ such
that for all $i\geq i_{0}$,

$\sup_{x\in K}|f_{i}(x)|<\epsilon.$

Now,

$\displaystyle\Big{|}\int
f_{i}\mathop{}\\!\mathrm{d}\mu\Big{|}\leq\int|f_{i}|\mathop{}\\!\mathrm{d}|\tilde{\mu}|$
$\displaystyle=\int_{K}|f_{i}|\mathop{}\\!\mathrm{d}|\tilde{\mu}|+\int_{X\mathbin{\backslash}K}|f_{i}|\mathop{}\\!\mathrm{d}|\tilde{\mu}|$
$\displaystyle\leq\sup_{x\in
K}|f_{i}(x)||\tilde{\mu}|(K)+C|\tilde{\mu}|(X\mathbin{\backslash}K)$
$\displaystyle<(|\mu|(X)+C)\,\epsilon.$

This shows convergence and, in conclusion, that $\mu$ is $k$-pre-regular. Now,
if $X$ is completely regular $\sigma$-compact $hk$-space or a Polish space,
then every Baire measure admits a unique extension to a Radon measure (see
4.2.1.3), so that indeed, every Baire measure is $k$-pre-regular. ∎

####  4.2.4. $k$-Regular measures

The space of $k$-pre-regular measures is still too large – it corresponds to
$C_{b}(X)^{\wedge}$ instead of the possibly smaller paired linear $hk$-space
dual $C_{b}(X)^{*}=\mathcal{M}(X)$. As a first step, the notion of _$k$
-regular measure_ solves this problem trivially by _requiring_ integration
against a $k$-regular measure to define a functional in $\mathcal{M}(X)$ (so
that the question becomes finding classes of examples of $k$-regular measures
that include the practically relevant ones).

######  Definition 4.2.4.1.

Let $X$ be an $hk$-space. A _$k$ -regular measure_ on $X$ is a $k$-pre-regular
measure $\mu$ such that,

$\int(-)\mathop{}\\!\mathrm{d}\mu\in C_{b}(X)^{*}(=\mathcal{M}(X)).$

As a consequence of this definition and 4.2.3.2, we may identify $k$-regular
measures with the elements of $\mathcal{M}(X)$ and will therefore use the same
notation $\mathcal{M}(X)$ also for the set of $k$-regular measures on $X$.
Similarly, we will write

$\mathcal{P}(X):=\\{\mu\in\mathcal{M}(X)\mid\mu\geq 0,\>\mu(1)=1\\}$

for the closed subspace of $\mathcal{M}(X)$ consisting of positive, normalised
functionals. In other words, $\mathcal{P}(X)$ is, under the identification of
measures and functionals, the space of $k$-regular probability measures on
$X$.

The pressing question is now, of course, whether the class of $k$-regular
measures is wide enough to encompass those examples which appear in
applications. As we will see, arbitrary Baire probability measures on compact
Hausdorff spaces are $k$-regular, and the same result holds for Polish spaces.
This will require examining the question of convergence in $\mathcal{M}(X)$
and how it relates to the more familiar notion of _weak convergence_ of
measures.

####  4.2.5. Convergence in spaces of $k$-regular measures

##### Weak convergence of measures

The following should be a familiar definition:

######  Definition 4.2.5.1.

Recall that a net $(\mu_{i})$ of measures on a topological space $X$ is said
to _converge weakly_ to $\mu$ if for all $f\in C_{b}(X)$,

$\int f\mathop{}\\!\mathrm{d}\mu_{i}\to\int f\mathop{}\\!\mathrm{d}\mu.$

When $(\mu_{i})$ is the sequence of distributions $(P_{x_{i}})$ of some random
variables $(x_{i})$, weak convergence of $(P_{x_{i}})$ is exactly _convergence
in distribution_ of the $(x_{i})$ in the sense of probability theory.

The simplest case in which convergence of _sequences_ in $\mathcal{M}(X)$ (and
of general nets in $\mathcal{P}(X)$) coincides with weak convergence is when
$X$ is compact. In this case, the notion of $k$-regular measure reduces to
that of Baire measure.

######  Proposition 4.2.5.2.

Let $X$ be a compact Hausdorff space. Then every Baire measure on $X$ is
$k$-regular (so that $\mathcal{M}(X)$ consists exactly of the Baire measures
on $X$). Moreover, the topology of $\mathcal{P}(X)$ coincides with the
topology of weak convergence of measures, and $\mathcal{P}(X)$ is a compact
Hausdorff space. In particular, a net of measures converges in
$\mathcal{M}(X)$ if, and only if, it converges weakly.

###### Proof.

When $X$ is compact, $C_{b}(X)=C(X)$ is the _Banach_ space of continuous
functions on $X$ and using Smith duality (2.3.0.1) together with 4.1.3.2, we
see that

$\mathcal{M}(X)=C(X)^{\wedge}.$

The left hand side is the space of $k$-regular measures, while the right hand
side can be identified with the space of Baire (equivalently, Radon, by
4.2.1.3) measures on $X$ (by the Riesz representation theorem, see [BR07,
Theorem 7.10.4]), so every Baire measure on $X$ is indeed $k$-regular. As a
closed subspace of the compact unit ball of the space
$\mathcal{M}(X)=C(X)^{\wedge}$, $\mathcal{P}(X)$ is compact and its topology
coincides with the weak-$*$ topology, i.e. the topology of weak convergence of
measures (see 2.2.3.4). ∎

##### Uniformly tight families and Prokhorov spaces

We will subsequently need the following measure-theoretic concepts.

######  Definition 4.2.5.3.

Recall that a family of Radon measures $(\mu_{i})$ on an $hk$-space $X$ is
_uniformly tight_ if for every $\epsilon>0$ there is a compact set $K$ such
that $|\mu|(X\mathbin{\backslash}K)<\epsilon$. $X$ is a _strongly sequentially
Prokhorov space_ if every weakly convergent sequence of Radon measures on $X$
is uniformly tight.

##### When does convergence in $\mathcal{P}(X)$ reduce to weak convergence?

######  Theorem 4.2.5.4.

Let $X$ be a either a Polish space or a completely regular hemicompact
$hk$-space. Then a sequence $(\mu_{n})$ of probability measures converges in
$\mathcal{P}(X)$ if, and only if, it converges weakly.

###### Proof.

Only the “if” direction is non-trivial, so suppose that $\mu_{n}\to\mu$
weakly. We want to show that $\mu_{n}\to\mu$ uniformly on compact subsets of
$C_{b}(X)$. So let $L\subseteq C_{b}(X)$ be compact and $\epsilon>0$. The
hypotheses imply that every Baire measure $\nu$ on $X$ extends uniquely to a
Radon measure $\tilde{\nu}$ (see 4.2.1.3) and that $X$ is strongly
sequentially Prokhorov (see [BR07, p. 220, Proposition 8.20.12] for the case
of hemicompact $hk$-spaces, and [BR07, p. 206, Theorem 8.6.8] for the case of
Polish spaces). Therefore, the family $\\{|\tilde{\mu}_{n}-\tilde{\mu}|\\}$ is
uniformly tight and there exists a compact subset $K_{\epsilon}\subseteq X$
such that for all $n\in\mathbb{N}$,
$|\tilde{\mu}_{n}-\tilde{\mu}|(X\mathbin{\backslash}K_{\epsilon})<\epsilon$.
Since the family of functions $L$ is compact, it is uniformly bounded by some
constant $C>0$. Moreover, given the hypotheses, the restrictions
$\tilde{\mu}_{n}|_{K_{\epsilon}},\tilde{\mu}|_{K_{\epsilon}}$ are Radon
measures on the compact Hausdorff space $K_{\epsilon}$, and
$\tilde{\mu}_{n}|_{K_{\epsilon}}\to\tilde{\mu}|_{K_{\epsilon}}$ weakly and
hence uniformly on compact subsets of $C(K_{\epsilon})$ (see 4.2.5.2). The
image of $L$ under restriction of functions to $K_{\epsilon}$,

$L|_{K_{\epsilon}}:=\\{f|_{K_{\epsilon}}\mid f\in L\\},$

is compact, since it is the image of the compact set $L$ under continuous map
$f\mapsto f\circ\iota_{K_{\epsilon}}$, where
$\iota_{K_{\epsilon}}:K_{\epsilon}\to X$ is the inclusion map. By compact
convergence of $\tilde{\mu}_{n}|_{K_{\epsilon}}$, we have that for
sufficiently large $n$,

$\sup_{f\in L|_{K_{\epsilon}}}\Big{|}\int
f\mathop{}\\!\mathrm{d}\tilde{\mu}_{n}|_{K_{\epsilon}}-\int
f\mathop{}\\!\mathrm{d}\tilde{\mu}|_{K_{\epsilon}}\Big{|}<\epsilon.$

Putting all of these observations together, we obtain that for all $f\in L$
and sufficiently large $n$,

$\displaystyle\Big{|}\int f\mathop{}\\!\mathrm{d}\mu_{n}-\int
f\mathop{}\\!\mathrm{d}\mu\Big{|}$
$\displaystyle\leq\int_{X\mathbin{\backslash}K_{\epsilon}}|f|\mathop{}\\!\mathrm{d}|\tilde{\mu}_{n}-\tilde{\mu}|+\Big{|}\int_{K_{\epsilon}}f\mathop{}\\!\mathrm{d}\tilde{\mu}_{n}-\int_{K_{\epsilon}}f\mathop{}\\!\mathrm{d}\tilde{\mu}_{n}\Big{|}$
$\displaystyle\leq
C\epsilon+\Big{|}\int_{K_{\epsilon}}f\mathop{}\\!\mathrm{d}\tilde{\mu}_{n}|_{K_{\epsilon}}-\int_{K_{\epsilon}}f\mathop{}\\!\mathrm{d}\tilde{\mu}|_{K_{\epsilon}}\Big{|}$
$\displaystyle\leq(C+1)\,\epsilon,$

which is what we wanted to show. ∎

######  Theorem 4.2.5.5.

Let $V$ be a separable Fréchet space. Then the Borel and Baire
$\sigma$-algebras on the natural dual $V^{\wedge}$ coincide and every Baire
measure on $V^{\wedge}$ is a Radon measure and $k$-regular. Hence,
$\mathcal{P}(V^{\wedge})$ consists exactly of the Radon probability measures
on $V^{\wedge}$. Moreover, a sequence of probability measures in
$\mathcal{P}(V^{\wedge})$ converges if, and only if, it converges weakly.
Finally, $\mathcal{P}(V^{\wedge})$ is a QCB space (in particular, a sequential
space) and consequently, convergence of sequences of probability measures
completely characterises the topology of $\mathcal{P}(V^{\wedge})$.

###### Proof.

The natural dual $V^{\wedge}$ of $V$ is a separable Brauner space and
therefore in particular a hemicompact QCB space. Since $V^{\wedge}$ is a
topological vector space, it is also completely regular and by 4.2.1.3, every
Baire measure has a unique extension to a Radon measure. Furthermore, since
$V^{\wedge}$ is a QCB space, it is hereditarily Lindelöf (see 2.1.6.5), so by
[BR07, p. 13, Corollary 6.3.5], the Baire and Borel sigma algebras coincide.
Therefore, every Baire measure on $V^{\wedge}$ is a Radon measure.

By 4.2.3.4, every Baire measure on $V^{\wedge}$ is a $k$-preregular measure.
To show that, moreover, every Baire (equivalently, Radon) measure on
$V^{\wedge}$ is $k$-regular, it suffices to show that for every Radon measure
$\mu$ on $V^{\wedge}$, there is a sequence of finitely supported measures
converging to $\mu$ in
$\mathcal{P}_{\mathrm{pre}}(V^{\wedge})=C_{b}(V^{\wedge})^{\wedge}$. Since, by
4.2.5.4, convergence in $\mathcal{P}(V^{\wedge})$ is equivalent to weak
convergence and
$\mathcal{P}(V^{\wedge})\subseteq\mathcal{P}_{\mathrm{pre}}(V^{\wedge})$ is
closed, it even suffices to show that for every Radon measure $\mu$ on
$V^{\wedge}$ there is a sequence of finitely supported measures converging
weakly to $\mu$. So let $\mu$ be a Radon measure on $V^{\wedge}$.

Let $K_{n}\subseteq V^{\wedge}$ be an increasing sequence of compact subsets
such that

$V^{\wedge}=\bigcup_{n\in\mathbb{N}}K_{n}.$

For each $n\in\mathbb{N}$, the restriction $\mu|_{K_{n}}$ is a Radon measure
on $K_{n}$. By 4.2.5.2, there exists a finitely supported measure $\nu_{n}$ on
$K_{n}$ such that for all $f\in C_{b}(V^{\wedge})$,

$\left|\int_{K_{n}}f\>\mathop{}\\!\mathrm{d}(\nu_{n}-\mu)\right|<\frac{1}{n}.$

We claim that the sequence $(\nu_{n})$ converges weakly to $\mu$. Let
$\epsilon>0$ and choose $N\in\mathbb{N}$ such that
$|\mu|(V^{\wedge}\mathbin{\backslash}K_{n})<\epsilon$ and $1/N<\epsilon$. Then
for all $n\geq N$,

$\displaystyle\left|\int_{V^{\wedge}}f\>\mathop{}\\!\mathrm{d}(\nu_{n}-\mu)\right|$
$\displaystyle\leq\left|\int_{V^{\wedge}\mathbin{\backslash}K_{n}}f\>\mathop{}\\!\mathrm{d}(\nu_{n}-\mu)\right|+\left|\int_{K_{n}}f\>\mathop{}\\!\mathrm{d}(\nu_{n}-\mu)\right|$
$\displaystyle<\|f\|_{\sup}\,\epsilon+\epsilon,$

proving the claim that $\nu_{n}$ converges weakly to $\mu$, which is hence a
$k$-regular measure.

Finally, the fact that $\mathcal{P}(V^{\wedge})$ is a QCB space follows from
the fact that QCB spaces are closed under the formation of countable
(co-)limits and exponentials (i.e. spaces of continuous maps). ∎

####  4.2.6. The Bochner-Minlos-Lévy-Fernique homeomorphism

Let us point out that, in conjunction with cartesian closure and sequentiality
of QCB spaces, 4.2.5.5 allows for a very concise simultaneous formulation of
the Bochner-Minlos and Lévy-Fernique theorems: if $V$ is a separable nuclear
Fréchet space, then the Fourier transform is a homeomorphism between
$\mathcal{P}(V^{\wedge})$ and the space of continuous normalised positive-
definite functions on $V$. Let us make this more precise.

######  Definition 4.2.6.1.

Let $V$ be a linear $hk$-space. Following [BR07, Definition 7.13.3], a
continuous function $f:V\to\mathbb{C}$ is _positive-definite_ if for all
$n\in\mathbb{N}$, $c_{1},\dots,c_{n}\in\mathbb{C}$, $x_{1},\dots,x_{n}\in V$,

$\sum_{i,j=1}^{n}c_{i}\overline{c_{j}}f(x_{i}-x_{j})\geq 0.$

We say that $f$ is _normalised_ if $f(0)=1$. Finally, we denote the set of
continuous normalised positive-definite functions on $V$ by $C_{1}^{+}(V)$ and
endow $C_{1}^{+}(V)$ with the subspace topology that it inherits from $C(V)$
as a closed subspace.

######  Definition 4.2.6.2.

Let $V$ be a linear $hk$-space and let $\mu\in\mathcal{P}(V^{\wedge})$ be a
probability measure. The _characteristic function_ of $\mu$ is the function
$\widehat{\mu}:V\to\mathbb{C}$ given by,

$\widehat{\mu}(x):=\int_{V^{\wedge}}e^{2\pi
i\phi(x)}\mathop{}\\!\mathrm{d}\mu(\phi).\qquad(x\in V)$

The map $\mu\mapsto\widehat{\mu}$ is called the _Fourier transform_.

With these definitions in place, we may formulate:

######  Theorem 4.2.6.3.

Let $V$ be a separable nuclear Fréchet space (for example,
$V=\mathcal{S}(\mathbb{R}^{n})$). Then the Fourier transform defines a
_homeomorphism_ ,

$\mathcal{F}:\mathcal{P}(V^{\wedge})\to
C_{1}^{+}(V),\;\;\mu\mapsto\int_{V^{\wedge}}e^{2\pi
i\phi(x)}\mathop{}\\!\mathrm{d}\mu(\phi).$

###### Proof.

The statement that $\mathcal{F}$ is a well-defined bijection is (a special
case of) the Bochner-Minlos theorem (see [Mey, Théorème 1]), taking into
account that by 4.2.5.5, $\mathcal{P}(V^{\wedge})$ consists exactly of the
Radon measures on $V^{\wedge}$. That $\mathcal{F}$ is continuous follows from
the fact that integration of continuous bounded functions against $k$-regular
measure is continuous (by definition) and the cartesian closure of
$hk\mathsf{Top}$. Finally, as a closed subspace of a QCB space $C(V)$,
$C_{1}^{+}(V)$ is a QCB space (see 2.1.6.4). In particular, $C_{1}^{+}(V)$ is
a sequential space. Therefore, the continuity of the inverse Fourier transform
$\mathcal{F}^{-1}$ is equivalent to its sequential continuity, and this
follows from the Lévy-Fernique continuity theorem ([Mey, Théorème 2]). ∎

######  Remark 4.2.6.4.

[Mey, Théorème 2] also entails that a sequence converges in $C_{1}^{+}(V)$ if,
and only if, it converges pointwise to a function that is continuous at the
origin.

####  4.2.7. $k$-Regular measures on Polish spaces

Our final observation concerning $k$-regular measures is that on Polish
spaces, their theory completely reduces to the classical one.

######  Proposition 4.2.7.1.

Let $X$ be a Polish space. Then following _every_ Baire measure is a
$k$-regular measure and hence, the notions of Baire, Radon and $k$-regular
measure all coincide (see 4.2.1.3). Moreover, the topology on $\mathcal{P}(X)$
is given by the topology of weak convergence of measures and $\mathcal{P}(X)$
is again a Polish space.

###### Proof.

Let $\mu$ be a Baire (equivalently, Radon, see 4.2.1.3) probability measure on
$X$. By 4.2.3.4, $\mu$ is $k$-pre-regular. Now, let $\mathcal{P}_{w}(X)$ be
the space of Radon measures with the topology of weak convergence, and recall
that $\mathcal{P}_{\mathrm{pre}}(X)\subseteq\mathcal{M}_{\mathrm{pre}}(X)$ is
the space of $k$-pre-regular probability measures with the subspace topology.
As sets, $\mathcal{P}_{w}(X)=\mathcal{P}_{\mathrm{pre}}$ (again, by 4.2.3.4).
We want to show that the respective topologies coincide, as well. First, note
that the identity map $\mathcal{P}_{\mathrm{pre}}(X)\to\mathcal{P}_{w}(X)$ is
continuous, since a convergent net in $\mathcal{P}(X)$ is also weakly
convergent. By [BR07, p. 213, Theorem 8.9.4], our assumption that $X$ is a
Polish space implies that $\mathcal{P}_{w}(X)$ is a Polish space, too. Hence,
sequential continuity of the identity map
$\mathcal{P}_{w}(X)\to\mathcal{P}_{\mathrm{pre}}(X)$ (which is provided by
4.2.5.4) implies continuity, so the two topologies coincide and
$\mathcal{P}_{\mathrm{pre}}(X)=\mathcal{P}_{w}(X)$ also as topological spaces.
Finally, finitely supported measures are dense in $\mathcal{P}_{w}(X)$ (see
[BR07, p. 213, Theorem 8.9.4 (ii)]) and therefore

$\mathcal{P}(X)=\mathcal{P}_{\mathrm{pre}}(X)=\mathcal{P}_{w}(X),$

which completes the proof. ∎

The above guarantees that most probability measures that occur in applications
are $k$-regular. (For a concrete example, take the law of a standard Brownian
motion, understood as Baire probability measure on the separable Fréchet space
$C([0,\infty),\mathbb{R}^{n})$.)

###  4.3. $\mathcal{M}$ as a commutative monad on $hk\mathsf{Top}$

####  4.3.1. The monad structure of $\mathcal{M}$

We now describe the monad structure of $\mathcal{M}$. Unit and multiplication
are given as follows.

######  Definition 4.3.1.1.

Define the (families of) map(s),

$\delta_{\bullet}:X\mapsto\mathcal{M}(X),\;\;x\mapsto\delta_{x},\;\;\;\;\;(X\in
hk\mathsf{Top})$

and

$\smallint{}:\mathcal{M}(\mathcal{M}(X))\to\mathcal{M}(X),\smallint{}(\pi)(f):=\int_{\mathcal{M(X)}}\mu(f)\mathop{}\\!\mathrm{d}\pi(\mu).\;\;\;\;\;(X\in
hk\mathsf{Top})$

Note that under the identification of measures with functionals, we have that

(4.1)
$\smallint{}(\pi)(f)=\pi(\mu\mapsto\mu(f)).\qquad(\pi\in\mathcal{M}(\mathcal{M}(X)),f\in
C_{b}(X))$

######  Lemma 4.3.1.2.

The families of maps $\delta_{\bullet}$ and $\smallint{}$ constitute well-
defined natural transformations.

###### Proof.

  1. 1.

_The map $\delta_{\bullet}$ is well-defined and continuous:_ This was 4.1.3.1.

  2. 2.

_Naturality of $\delta_{\bullet}$:_ We need to show that for all $hk$-spaces
$X,Y$ and every continuous map $f:X\to Y$ the following diagram commutes:

${X}$${Y}$${{\mathcal{M}(X)}}$${{\mathcal{M}(Y)}}$$\scriptstyle{f}$$\scriptstyle{\delta_{\bullet}}$$\scriptstyle{\delta_{\bullet}}$$\scriptstyle{f_{*}}$

But this just means that for all $x\in X$,

$f_{*}\delta_{x}=\delta_{f(x)},$

which holds.

  3. 3.

_The map $\smallint{}$ is well-defined and continuous:_ First notice that
$\smallint{}$ is well-defined, since for every $f\in C_{b}(X)$, the map
$\mu\mapsto\mu(f)$ is continuous and bounded (by $\|f\|_{\sup}\cdot|\mu|(X)$)
on $\mathcal{M}(X)$. Moreover, being given by a certain evaluation-type map
(see Eq. 4.1), $\smallint{}$ is continuous by cartesian closure of
$hk\mathsf{Top}$.

  4. 4.

_The map $\smallint{}$ is natural:_ We need to show that the following diagram
commutes:

${{\mathcal{M}(\mathcal{M}(X))}}$${{\mathcal{M}(\mathcal{M}(Y))}}$${{\mathcal{M}(X)}}$${{\mathcal{M}(Y)}}$$\scriptstyle{(f_{*})_{*}}$$\scriptstyle{\smallint{}}$$\scriptstyle{\smallint{}}$$\scriptstyle{f_{*}}$

To prove this, we calculate, using Eq. 4.1, that for all
$\pi\in\mathcal{M}(\mathcal{M}(X))$ and all $h\in C_{b}(X)$,

$\displaystyle f_{*}\left(\smallint{}(\pi)\right)(h)$
$\displaystyle=\smallint{}(\pi)(h\circ f)$
$\displaystyle=\pi(\mu\mapsto\mu(h\circ f))$ $\displaystyle=\pi(\mu\mapsto
f_{*}\mu(h))$ $\displaystyle=\pi(f_{*}\circ(\mu\mapsto\mu(h)))$
$\displaystyle=(f_{*})_{*}\pi(\mu\mapsto\mu(h))$
$\displaystyle=\smallint{}((f_{*})_{*}\pi)(h).$

∎

######  Theorem 4.3.1.3.

$(\mathcal{M},\delta_{\bullet},\smallint{})$ is a monad on $hk\mathsf{Top}$.

###### Proof.

We need to verify the monad laws, which amounts to the following three
calculations, where $X$ is an $hk$-space, $\mu\in\mathcal{M}(X)$, $f\in
C_{b}(X)$ and $\pi\in\mathcal{M}(\mathcal{M}(X))$.

  1. 1.

First unit law:

$\smallint{}(\delta_{\mu})(f)=\int\nu(f)\mathop{}\\!\mathrm{d}\delta_{\mu}(\nu)=\mu(f).$

  2. 2.

Second unit law:

$(\smallint{}(\delta_{\bullet})_{*}\mu)(f)=\int\delta_{x}(f)\mathop{}\\!\mathrm{d}\mu=\int
f\mathop{}\\!\mathrm{d}\mu=\mu(f)$

  3. 3.

Associativity law: We need to show that for all
$\Pi\in\mathcal{M}(\mathcal{M}(\mathcal{M}(X)))$ and all $h\in C_{b}(X)$,

$\smallint{}\left(\smallint{}(\Pi)\right)(h)=\smallint{}\left(\smallint{}\right)_{*}\Pi.$

To do so, we repeatedly use Eq. 4.1 to obtain,

$\displaystyle\smallint{}\left(\smallint{}(\Pi)\right)(h)$
$\displaystyle=\left(\smallint{}(\Pi)\right)(\mu\mapsto\mu(h))$
$\displaystyle=\Pi(\nu\mapsto\nu(\mu\mapsto\mu(h)))$
$\displaystyle=\Pi(\nu\mapsto\left(\smallint{}(\nu)\right)(h))$
$\displaystyle=\Pi\left([\mu\mapsto\mu(h)]\circ\smallint{}\right)$
$\displaystyle=\left(\smallint{}\right)_{*}\Pi(\mu\mapsto\mu(h))$
$\displaystyle=\smallint{}\left(\smallint{}\right)_{*}\Pi,$

completing the proof.

∎

####  4.3.2. Product measures (the commutative monad structure of
$\mathcal{M}$)

######  Definition 4.3.2.1.

Let $X,Y$ be $hk$-spaces and let $\mu\in\mathcal{M}(X)$,
$\nu\in\mathcal{M}(Y)$ be $k$-regular measures. Define the product measure
$\mu\otimes\nu\in\mathcal{M}(X\times Y)$ on $X\times Y$ as (the $k$-regular
measure representing the functional),

$(\mu\otimes\nu)(f):=\int\int
f(x,y)\mathop{}\\!\mathrm{d}\mu(x)\mathop{}\\!\mathrm{d}\nu(y).\;\;\;\;\;(f\in
C_{b}(X\times Y))$

The family of maps

$\otimes:\mathcal{M}(X)\times\mathcal{M}(Y)\to\mathcal{M}(X\times
Y),\;\;(\mu,\nu)\mapsto\mu\otimes\nu,\;\;\;\;((X,Y)\in hk\mathsf{Top}\times
hk\mathsf{Top})$

is now a natural transformation (this follows from a change of variables in
the definition) and we have that:

######  Theorem 4.3.2.2.

$(\mathcal{M},\delta_{\bullet},\smallint{},\otimes)$ is a commutative monad on
$hk\mathsf{Top}$.

###### Proof.

We need to show that the diagram

${{\mathcal{M}(X)\times\mathcal{M}(Y)}}$${{\mathcal{M}(X\times
Y)}}$${{\mathcal{M}(Y)\times\mathcal{M}(X)}}$${{\mathcal{M}(Y\times
X)}}$$\scriptstyle{\otimes}$$\scriptstyle{\sigma}$$\scriptstyle{\otimes}$$\scriptstyle{\sigma_{*}}$

commutes (where $\sigma$ is the braiding), which amounts to showing that for
all $f\in C_{b}(X)$,

$\int f\mathop{}\\!\mathrm{d}\mu\otimes\nu=\int
f\mathop{}\\!\mathrm{d}\sigma_{*}(\nu\otimes\mu).$

After a change of variables and expanding, this becomes,

(4.2) $\int\int
f(x,y)\mathop{}\\!\mathrm{d}\mu(x)\mathop{}\\!\mathrm{d}\nu(y)=\int\int
f(x,y)\mathop{}\\!\mathrm{d}\nu(y)\mathop{}\\!\mathrm{d}\mu(x).$

We might now be tempted to invoke a version of Fubini’s theorem for signed
measures, but let us take the oppurtunity to give an independent proof which
will then instead _entail_ this version of Fubini’s theorem.

Note that both sides of (4.2) are continuous in $(\mu,\nu)$ and that equality
holds when $\mu$ and $\nu$ are finitely supported (since then, (4.2) reduces
to an interchange of finite sums). The claim now follows from density of
finitely supported measures in $\mathcal{M}(X\times Y)$. ∎

###  4.4. The Probability Monad $\mathcal{P}$ on the Category of $hk$-Spaces

Because the pushforward of a probability measure remains a probability
measure, $\mathcal{P}(-)$ inherits the structure of a functor from
$\mathcal{M}$. Similarly, we can equip $\mathcal{P}$ with a unit and
multiplication coming from the monad structure of $\mathcal{M}$.

######  Definition 4.4.0.1.

Define the family of continuous maps,

$\delta_{\bullet}:X\mapsto\mathcal{P}(X),\;\;x\mapsto\delta_{x},\;\;\;\;\;(X\in
hk\mathsf{Top})$

as well as,

$\smallint{}:\mathcal{P}(\mathcal{P}(X))\to\mathcal{P}(X),\smallint{}(\pi)(f):=\int_{\mathcal{P}(X)}\mu(f)\mathop{}\\!\mathrm{d}\pi(\mu).\;\;\;\;\;(X\in
hk\mathsf{Top})$

Note that this definition is exactly the same as 4.3.1.1 if we replace
$\mathcal{P}$ with $\mathcal{M}$. Moreover, when $\mu$ and $\nu$ probability
measures then so is the product measure $\mu\otimes\nu$ from 4.3.2.1 and we
obtain a family of maps

$\otimes:\mathcal{P}(X)\times\mathcal{P}(Y)\to\mathcal{P}(X\times
Y),\;\;(\mu,\nu)\mapsto\mu\otimes\nu,\;\;\;\;((X,Y)\in hk\mathsf{Top}\times
hk\mathsf{Top})$

We can therefore, the proof being exactly as before, conclude:

######  Theorem 4.4.0.2.

The families $\delta_{\bullet}$, $\smallint{}$ and $\otimes$ constitute
natural transformations and
$(\mathcal{P},\delta_{\bullet},\smallint{},\otimes)$ is a commutative monad on
$hk\mathsf{Top}$.

######  Remark 4.4.0.3.

Now that we know that $\mathcal{P}$ is a monad, we may rephrase the results of
4.2.5.2 and 4.2.7.1 as saying that $\mathcal{P}$ restricts to the Giry monad
on Polish spaces, and to the Radon monad on compact Hausdorff spaces,
recovering (and in some sense “interpolating” between) two well-known
probability monads. Moreover, since the category $h\mathsf{QCB}$ of
$k$-Hausdorff QCB spaces is closed under the formation of limits and
exponentials in $hk\mathsf{Top}$, $\mathcal{P}$ also restricts to a
probability monad on $h\mathsf{QCB}$.

###  4.5. The Free Paired Linear $hk$-Space $\mathcal{M}_{c}(X)$ as a Space of
Compactly Supported Measures

In Chapter 3, we defined $\mathcal{M}_{c}(X)$ to be the free paired linear
$hk$-space (equivalently, the free replete linear $hk$-space) on $X$ (see
3.3.5.15). We now justify this notation by showing that $\mathcal{M}_{c}(X)$
can be identified with with the space of compactly supported $k$-regular
measures on $X$. In contrast to $\mathcal{M}(X)$, of which we needed to show
“by hand” that it defines a commutative monad, this is automatic for
$\mathcal{M}_{c}(X)$ from the general considerations of Chapter 3 (see Remark
3.3.5.17).

Let us start by observing that we may regard $\mathcal{M}_{c}(X)$ as a subset
of $\mathcal{M}(X)$.

######  Lemma 4.5.0.1.

The map

$i^{*}:\mathcal{M}_{c}(X)\to\mathcal{M}(X)$

obtained as the adjoint of the inclusion map $i:C_{b}(X)\to C(X)$ is
injective.

###### Proof.

Let $\mu,\nu\in\mathcal{M}_{c}(X)$ and suppose that $i^{*}(\mu)=i^{*}(\nu)$.
Then, for all $f\in C_{b}(X)$, $\nu(f)=\mu(f)$. Since $C_{b}(X)$ is dense in
$C(X)$, this implies that $\mu=\nu$. ∎

######  Definition 4.5.0.2.

Let $X$ be an $hk$-space. By a compactly supported $k$-regular measure on $X$
we mean a $k$-regular measure $\mu\in\widetilde{\mathcal{M}}(X)$ which is the
pushforward of a Baire measure defined on a compact subspace of $X$ under the
inclusion. (Of course, by 4.2.5.2, $k$-regularity of $\mu$ is automatic in
this case.) We denote the set of compactly supported $k$-regular measures by
$\widetilde{\mathcal{M}}_{c}(X)$.

######  Remark 4.5.0.3.

Why do we not just define compactly supported $k$-regular measures as
“$k$-regular measures whose support is compact”? This is for a technical
reason: Remember that on a general topological space, the Baire
$\sigma$-algebra may be a proper subset of the Borel $\sigma$-algebra, which
is why the usual definition of the support of a _Radon_ (or, more generally,
_$\tau$ -additive_) measure cannot be applied here.

However, if $X$ is a _Hausdorff_ $hk$-space, then every tight (in particular,
every compactly supported $k$-regular) Baire measure on $X$ admits a unique
extension to a Radon measure [BR07, Theorem 7.3.10]. Hence, for all practical
purposes, the set of compactly supported $k$-regular measures can be
identified with the set of compactly supported Radon measures.

Before we can provide an identification between compactly supported measures
in $\widetilde{\mathcal{M}}_{c}(X)$ and functionals in $\mathcal{M}_{c}(X)$,
as promised, we need a lemma.

######  Lemma 4.5.0.4.

Let $X$ be an $hk$-space and let $\phi$ be a continuous linear functional on
the space $C_{\mathsf{c.o.}}(X)$ of continuous functions _with the compact-
open topology_. Then there exists a compactly supported $k$-regular measure
$\mu$ on $X$ such that for all $f\in C(X)$,

$\phi(f)=\int_{X}f(x)\mathop{}\\!\mathrm{d}\mu(x).$

###### Proof.

Put

$P:=\prod_{K\subseteq X\text{
compact}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{}^{\mathsf{Top}}\;\;C(K),$

where $\prod^{\mathsf{Top}}$ denotes the product in $\mathsf{Top}$
(equivalently, in the category of locally convex topological vector spaces).
Since $X$ is $k$-Hausdorff, the compact-open topology is the topology of
uniform convergence on compact subsets and hence, the map

$C_{\mathsf{c.o.}}(X)\hookrightarrow P,\;\;\;f\mapsto(f|_{K})$

is an embedding of locally convex topological vector spaces. By the Hahn-
Banach extension theorem, $\phi$ has an extension $\tilde{\phi}$ to $P$. Every
linear functional on a product of locally convex topological vector spaces is
a finite sum of functionals factoring through one of its factors, so there
exist $K_{1},\dots,K_{n}\subseteq X$ compact and $\phi_{1}\in
C(K_{1})^{\wedge},\dots\phi_{n}\in C(K_{n})^{\wedge}$ such that

$\tilde{\phi}=\sum_{i=1}^{n}\phi_{i}\circ\pi_{i},$

where $\pi_{i}:P\to C(K_{i})$ is the canonical projection. By the
Riesz–Markov–Kakutani theorem for the case of compact Hausdorff spaces, there
exist Baire measures $\mu_{1},\dots,\mu_{n}$ on $K_{1},\dots,K_{n}$ such that

$\phi_{i}=\int_{K_{i}}(-)\mathop{}\\!\mathrm{d}\mu_{i},$

for every $i\in\\{1,\dots,n\\}$. Hence, we obtain that

$\phi=\sum_{i=1}^{n}\int_{K_{i}}(-)\mathop{}\\!\mathrm{d}\mu_{i}.$

Let $\iota_{i}:K_{i}\hookrightarrow X$ be the inclusion and let

$\mu:=\sum_{i=1}^{n}(\iota_{i})_{*}\mu_{i}\in\mathcal{M}_{c}(X).$

Then we have that

$\phi=\int_{X}(-)\mathop{}\\!\mathrm{d}\mu,$

which is what we wanted to show. ∎

######  Theorem 4.5.0.5.

Let $X$ be an $hk$-space. Then the map

$\Phi:\widetilde{\mathcal{M}}_{c}(X)\to\mathcal{M}_{c}(X),\;\;\;\mu\mapsto\int_{X}(-)\mathop{}\\!\mathrm{d}\mu$

is a well-defined natural bijection between the set
$\widetilde{\mathcal{M}}_{c}(X)$ of compactly supported $k$-regular measures
on $X$ and the set $C(X)^{*}\cong\mathcal{M}_{c}(X)$. In particular, when $X$
is Hausdorff $\mathcal{M}_{c}(X)$ can be identified with the set of compactly
supported Radon measures on $X$ (see Remark 4.5.0.3).

###### Proof.

  1. 1.

_$\Phi$ is well-defined:_

    1. 1.1.

_$\Phi(\mu)$ is continuous on $C(X)$ (for any
$\mu\in\widetilde{\mathcal{M}}_{c}(X)$:_ it is even continuous on
$C_{\mathsf{c.o.}}(X)$, since

$\Big{|}\int_{X}f(x)\mathop{}\\!\mathrm{d}\mu(x)\Big{|}\leq\mu(X)\,\sup_{x\in
K}|f(x)|,$

for all $f\in C(X)$, where $K:=\operatorname*{supp}\mu$ is the support of
$\mu$.

    2. 1.2.

_$\operatorname*{im}\Phi\subseteq\mathcal{M}_{c}(X)$ :_ Let
$\mu\in\widetilde{\mathcal{M}}_{c}(X)$. We want to show that

$\tilde{\mu}:=\int_{X}(-)\mathop{}\\!\mathrm{d}\mu\in\mathcal{M}_{c}(X).$

Let $K:=\operatorname*{supp}\mu$. Since $K$ is compact,
$\mathcal{M}_{c}(K)=C(K)^{\wedge}$. Now,

$\tilde{\mu}|_{K}:=\int_{K}(-)\mathop{}\\!\mathrm{d}\mu\in\mathcal{M}_{c}(K)=C(K)^{\wedge}.$

Let $\iota:K\hookrightarrow X$ be the inclusion. Then

$\tilde{\mu}=\iota_{*}(\tilde{\mu}|_{K})\in\mathcal{M}_{c}(X).$

  2. 2.

_$\Phi$ is injective:_ Since $\mathcal{M}_{c}(X)$ can be identified with a
subspace of $\mathcal{M}(X)$ (4.5.0.1), we already know that $\Phi$ is
injective from 4.2.2.1.

  3. 3.

_$\Phi$ is surjective:_ As $\Phi$ has dense image (since linear combinations
of point masses are dense in $\mathcal{M}_{c}(X)$), it suffices to show that
$\operatorname*{im}\Phi\subseteq\mathcal{M}_{c}(X)$ is closed.
$\mathcal{M}_{c}(X)$ is a closed subspace of $C(C(X))$, by definition, and a
closed subset of $C_{\mathsf{c.o.}}(C(X))$ is also closed in $C(C(X))$. It is
therefore sufficient to prove that $\operatorname*{im}\Phi$ is closed as a
subset of $C_{\mathsf{c.o.}}(C(X))$. Moreover,
$C_{\mathsf{c.o.}}(C_{\mathsf{c.o.}}(X))$ is a subspace (in the sense of
carrying the subspace topology) of $C_{\mathsf{c.o.}}(C(X))$ because the
compact subsets of $C(X)$ and $C_{\mathsf{c.o.}}(X)$ coincide. This in turn
reduces the claim to showing that $\operatorname*{im}\Phi$ is closed in
$C_{\mathsf{c.o.}}(C_{\mathsf{c.o.}}(X))$, which follows from the fact that
$\operatorname*{im}\Phi$ consists precisely of those continuous functions in
$C_{\mathsf{c.o.}}(C_{\mathsf{c.o.}}(X))$ which are linear, by 4.5.0.4.

∎

###  4.6. Monadic Vector-Valued Integration

####  4.6.1. The vector-valued integral

We now show how the preceding developments can be applied to obtain a
particularly simple and coherent theory of vector-valued integration.
Implementing a general idea of Kock [Koc11] which was refined and further
developed by Lucyshyn-Wright in his doctoral thesis [LW13] (see there for
further references), we realise our vector valued integral by viewing paired
linear $hk$-spaces as modules over the commutative monad $\mathcal{M}_{c}$.
All the desired properties of the resulting vector-valued integral follow
naturally from this, without much further work, which is the reason for the
term _monadic vector-valued integration_. In this setting, the two notational
components of the integral, the integral sign $\smallint{}$ and what comes
after it (“$f(x)\mathop{}\\!\mathrm{d}\mu(x)$”), both have separate precise
meanings which we now define.

######  Definition 4.6.1.1.

Let $V$ be a paired linear $hk$-space. Since $\mathcal{M}_{c}(V)$ is the free
paired linear $hk$-space on $V$, the identity $V\to V$ extends uniquely to a
morphism $\mathcal{M}_{c}(V)\to V$ of paired linear $hk$-spaces. We denote
this map by

$\smallint{}:\mathcal{M}_{c}(V)\to
V\;\;\;\;\;\text{or}\;\;\;\;\int:\mathcal{M}_{c}(V)\to V$

Moreover, let us introduce the following notation for the pushforward:

$f(x)\mathop{}\\!\mathrm{d}\mu(x):=f_{*}\mu.$

That this definition agrees with the usual Lebesgue integral in the scalar
case is guaranteed by the following fact.

######  Proposition 4.6.1.2.

Let $f:\mathbb{K}\to\mathbb{K}$ be continuous and let
$\mu\in\mathcal{M}_{c}(\mathbb{K})$. Then,

$\smallint{}(f_{*}\mu)=\int_{\mathbb{K}}f(x)\mathop{}\\!\mathrm{d}\mu(x)=\int_{\mathbb{K}}f(x)\mathop{}\\!\mathrm{d}\mu(x),$

where the right hand side denotes a Lebesgue integral in the usual sense, and
the $\smallint{}$ on the left hand side denotes the map,

$\smallint{}:\mathcal{M}_{c}(\mathbb{K})\to\mathbb{K},\;\;\nu\mapsto\nu(\mathsf{id}_{\mathbb{K}}),$

from 4.6.1.1.

###### Proof.

This is simply the change-of-variables formula for the Lebesgue integral:

$\int_{\mathbb{K}}f(x)\mathop{}\\!\mathrm{d}\mu(x)=\int_{\mathbb{K}}x\mathop{}\\!\mathrm{d}f_{*}\mu(x)=(f_{*}\mu)(\mathsf{id}_{\mathbb{K}})=\smallint{}(f_{*}\mu).$

∎

######  Remark 4.6.1.3.

We restrict our attention to integration of continuous maps against compactly
supported measures for two reasons. On the one hand, this is the setting which
naturally arises from the monad structure of $\mathcal{M}_{c}$, allowing us to
contemplate the phenomena specific to this monadic vector-valued integration
theory. On the other hand, this is a sufficient level of generality for many
important cases of vector-valued integration such as the integration of
continuous curves (which is certainly covered by this setting). For example,
integrals of continuous curves are crucial in infinite-dimensional calculus
(see Part I of [Ham82]), as well as the theory of vector-valued holomorphic
functions (see [Gar18, Chapter 15]).

######  Remark 4.6.1.4.

We can now interpret the ($hk\mathsf{Top}$-enriched) free-forgetful adjunction
between paired linear $hk$-spaces and $hk$-spaces (see Chapter 3, Remark