category theory

mapping space

# Contents

## Idea

For $\mathcal{C}$ a category and $X, Y \in \mathcal{C}$ two objects, the internal hom $[X,Y] \in \mathcal{C}$ from $X$ to $Y$ is, if it exists, another object of $\mathcal{C}$ which behaves like the “object of morphisms” from $X$ to $Y$. In other words it is, if it exists, an internal version of the ordinary hom set $\mathcal{C} \in Set$ or more generally hom object $\mathcal{C}(X, Y) \in \mathcal{V}$ of a locally small category or $\mathcal{V}$-enriched category.

One way to make this precise is to mimic the basic property of a function set $[X,Y] = \{f : X \to Y\}$ of functions between two sets $X$ and $Y$: that is uniquely characterizes by the fact that for any other set $S$ the functions $S \to [X,Y]$ are in natural bijection with the functions $S \times X \to Y$ out of the cartesian product of $S$ with $X$. Formally this says that the functor $X\times (-)$ of taking the cartesian product with the set $S$ has a right adjoint given by the construction $[X,-]$.

This, then, is, generally, the definition of internal hom in any cartesian monoidal category or in fact in any monoidal category $(\mathcal{C}, \otimes)$: the right adjoint $[X,-]$ to the given tensor product functor $(-)\otimes X$ for all objects $X$. It may or may not exist. If it exists, one says that $(\mathcal{C}, \otimes)$ is a closed monoidal category. Explicity, the condition is that there is an isomorphism(bijection)

$\mathcal{C}(A, [X,Z]) \stackrel{\simeq}{\to} \mathcal{C}(A \otimes X, Z)$

which is natural in all three variables. (The rightward map here is often called currying, especially in a closed monoidal category (and more especially for the $\lambda$-calculus).)

In particular this implies that in a closed monoidal category the external hom is re-obtained from the internal hom as its set of generalized elements out of the tensor unit $I \in \mathcal{C}$ in that

$\frac{I \to [X,Y]}{X \to Y}$

using that $I \otimes X \simeq X$ by definition of the tensor unit.

Here “closed” in “closed monoidal category” is in the sense that forming “hom-sets” does not lead “out of the category”. In fact the internal hom of a cartesian monoidal category is indeed the hom as seen in the internal logic of that category (the function type).

More generally, one can consider objects that satisfy some basic universal properties that an internal hom should satisfy even in the absence of a monoidal structure. If such objects exist one speaks therefore just of a closed category. Every closed category may be seen as a category enriched over itself. Accordingly, an internal hom is after all a special case of a hom-object, for the special case of this enrichment over itself.

## Definition

###### Definition

Let $(\mathcal{C}, \otimes)$ be a monoidal category. An internal hom in $\mathcal{C}$ is a functor

$[-,-] : \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}$

such that for every object $X \in \mathcal{C}$ we have a pair of adjoint functors

$((-) \otimes X \dashv [X, -]) : \mathcal{C} \to \mathcal{C} \,.$

If this exists, $(\mathcal{C}, \otimes)$ is called a closed monoidal category.

## Properties

### Evaluation map

Let $(\mathcal{C}, \otimes)$ be a closed monoidal category.

###### Definition

For $X,Y \in \mathcal{C}$ two objects, the evaluation map

$eval_{X,Y} : [X,Y] \otimes X \to Y$

is the $((-)\otimes X \vdash [X,-])$-adjunct of the identity $id_{[X,Y]} : [X,Y] \to [X,Y]$.

###### Remark

If $\mathcal{C}$ is specifically a locally cartesian closed category, then in terms of the type theory internal language of $\mathcal{C}$ the evaluation map is the categorical semantics of the dependent type which in type theory syntax is

$f \colon X \to Y,\; x \colon X \;\vdash\; f(x) \colon Y \,,$

with function application on the right.

### Composition map

Let $(\mathcal{C}, \times)$ be a closed monoidal category.

###### Definition

For $X, Y, Z \in \mathcal{C}$ three objects, the composition morphism

$\circ_{X,Y,Z} : [Y, Z] \times [X, Y] \to [X, Z]$

is the $((-)\times X \vdash [X,-])$-adjunct of the following composite of two evaluation maps, def. 2:

$[Y, Z] \times [X , Y] \times X \stackrel{(id_{[Y,Z]}, eval_{X,Y})}{\to} [Y,Z] \times Y \stackrel{eval_{Y,Z}}{\to} Z \,.$

### Relation to function types

The internal hom is the categorical semantics of what in type theory are function types

type theorycategory theory
syntaxsemantics
natural deductionuniversal construction
function typeinternal hom
type formation$\frac{\vdash\: X \colon Type \;\;\;\;\; \vdash\; A\colon Type}{\vdash \; \left(X \to A\right) \colon Type}$
term introduction$\frac{x \colon X \;\vdash\; a(x) \colon A}{\vdash (x \mapsto a\left(x\right)) \colon \left(X \to A\right) }$
term elimination$\frac{\vdash\; f \colon \left(X \to A\right)\;\;\;\; \vdash \; x \colon X}{x \colon X\;\vdash\; f(x) \colon A}$
computation rule$(y \mapsto a(y))(x) = a(x)$

For each object $S$ the (internal hom $\dashv$ tensor product)-adjunction induces a monad $[S, S \otimes (-)]$. In computer science this monad (in computer science) is called the state monad.

## Examples

### In $Set$

In the category Set of sets, regarded as a cartesian monoidal category, the internal hom is given by function sets. This exists, by the discussion there, as soon as the foundational axioms are strong enough, for instance as soon as there are power objects, which is the special case of a function set into the 2-element set.

### In a sheaf topos or $(\infty,1)$-sheaf $(\infty,1)$-topos

Let $C$ be a site. Let $\mathbf{H} = Sh(C)$ be the sheaf topos over $C$ or in fact the (∞,1)-sheaf (∞,1)-topos. We discuss the internal hom of this regard as a cartesian monoidal category/cartesian monoidal (∞,1)-category.

###### Proposition

The sheaf topos $\mathbf{H}$ is a cartesian closed category / cartesian closed (∞,1)-category. In fact it is a locally cartesian closed category / locally cartesian closed (∞,1)-category.

Hence the internal hom exist.

=–

###### Proposition

For $X, Y \in \mathbf{H}$ two objects, the internal hom-object

$[X,Y] \in \mathbf{H}$

is the sheaf/(∞,1)-sheaf given by the assignment

$[X,Y] : U \mapsto \mathbf{H}(U \times X, Y) \,,$

for all objects $U \in C$ which on the right we regard under the Yoneda embedding/∞-Yoneda embedding $U \in C \stackrel{Yoneda}{\hookrightarrow} \mathbf{H}$.

Here

• $U \times X \in \mathbf{H}$ is the cartesian product if $U$ with $X$

• $\mathbf{H}(-,-)$ is the hom set-functor / hom space-(∞,1)-functor of $\mathbf{H}$.

###### Proof

By the Yoneda lemma/(∞,1)-Yoneda lemma we have natural equivalences

$[X,Y](U) \simeq \mathbf{H}(U , [X,Y])$

and by the defining $((-)\times X \vdash [X,-])$adjunction this is naturally equivalent to

$\cdots \simeq \mathbf{H}(U \times X, Y) \,.$
###### Remark

In the (homotopy-)type theory syntax of the internal language of $\mathbf{H}$ the internal hom $[X, Y] \in \mathbf{H}$ is the categorical semantics of the function type

$\vdash (X \to Y) : Type \,.$
###### Proposition

For $X, Y \in \mathbf{H}$, the evaluation map, def. 2,

$eval_{X,Y} : [X,Y] \times X \to Y$

is the morphism of sheaves which over each $U \in C$ sends a morphism of sheaves $\theta : \mathbf{H}(-,U) \times X(-) \to Y(-)$ (which is the first component by prop. 2) and an $x \in \mathbf{H}(U,X)$ to

$eval_{X,Y}(U) : (\theta, x) \mapsto \theta_U(id_U, x) \in Y(U) \,.$

See (MacLane-Moerdijk, p. 46).

###### Proposition

For $X, Y, Z \in \mathbf{H}$ three objects of $\mathbf{H}$, the canonical composition morphism, def. 3,

$\circ_{X,Y,Z} : [Y, Z] \times [X, Y] \to [X, Z]$

is given by the morphism of presheaves/(∞,1)-presheaves whose component over $U \in C$ is the morphism of sets/∞-groupoids

$\circ_{X,Y,Z}(U) : \mathbf{H}(U \times X, Y) \times \mathbf{H}(U \times Y, Z) \to \mathbf{H}(U \times X, Z)$

which sends a pair $(f : U \times X \to Y, g : U \times Y \to Z)$ to the composite

$\circ_{X,Y,Z}(U)(f,g) = U \times X \stackrel{(\Delta_U, id_X)}{\to} U \times U \times X \stackrel{(id_U, f)}{\to} U \times Y \stackrel{g}{\to} Z \,,$

where $\Delta_U : U \to U \times U$ is the diagonal morphism on $U$.

###### Proof

By definition 3 the morphism is the adjunct of the double evaluation map

$[Y,Z] \times [X,Y] \times X \to Z \,.$

Since the cartesian product of two sheaves $A, B \in \mathbf{H}$ is computed objectwise

$A \times B : U \mapsto A(U) \times B(U)$

it follows that over each $U \in C$ this double evaluation map is the morphism of sets/$\infty$-groupoids

$[Y,Z](U) \times [X,Y](U) \times X(U) \to Z(U)$

hence by prop. 2

$\mathbf{H}(U \times Y, Z) \times \mathbf{H}(U \times X, Y) \times \mathbf{H}(U,X) \to \mathbf{H}(U,Z) \,,$

where now by prop. \ref this is the external evaluation.

###### Remark

Intuitively this says that the composite of a $U$-parameterized family of maps $\{f(u) : X \to Y| u \in U\}$ with a $U$-parameterized family of maps ${g(u) : Y \to Z| u \in U}$ is the $U$-family given by the parameter-wise composite $\{g(u)\circ f(u) | u \in U\}$.

###### Example

The internal automorphism group/automorphism ∞-group of an object $X \in \mathbf{H}$ is the subobject

$\mathbf{Aut}(X) \hookrightarrow [X,X]$

of the internal hom which is maximal subject to the property that the composition of prop. 4 becomes invertible.

The (homotopy-)type theory syntax for this is given by the type of equivalences in homotopy type theory

$\vdash (X \stackrel{\simeq}{\to} X) : Type \,.$

### In slice categories

Let $\mathbf{H}$ be a locally cartesian closed category. This means that for each object $X \in \mathbf{H}$ the slice category $\mathbf{H}_{/X}$ is a cartesian closed category. The product in the slice is given by the fiber product over $X$ computed in $\mathbf{H}$. Fairly detailed discussion of constructions of the internal hom in such slices $\mathbf{H}_{/X}$ is at locally cartesian closed category – cartesian closure in terms of base change and dependent product.

We record some further properties

###### Proposition

For $\mathcal{C}$ a locally cartesian closed category and $f \colon X \to Y$ any morphism in $\mathcal{C}$, the inverse image $f^*$ of the corresponding base change adjunction

$\mathcal{C}_{/X} \stackrel{\overset{\sum_f}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{\prod_f}{\to}}} \mathcal{C}_{/Y}$

This is discussed in more detail at cartesian closed functor – Examples.

So for $A,B \in \mathcal{C}_{/Y}$ we have isomorphisms

$f^* \left[A,B\right] \stackrel{\simeq}{\to} \left[f^* A , f^* B\right]$

between the image of the internal hom under $f^*$ and the internal hom of the images of $A$ and $B$ separately.

###### Proposition

For $\mathbf{H}$ a locally cartesian closed category, $f \colon X \to Y$ any morphism, and $A, B \in \mathbf{H}_{/X}$ two objects in the slice over $X$, there is a natural morphism

$\prod_f \left[A,B \right] \to \left[ \sum_f A, \sum_f B\right] \,.$

Here are two ways to get this morphism:

###### Proof/Construction 1

For any object $U \in \mathbf{H}_{/Y}$ we have a canonical morphism of hom sets

\begin{aligned} \mathbf{H}_{/Y}( U, \prod_f [A,B] ) & \simeq \mathbf{H}_{/X}( f^* U, [A,B] ) \\ & \simeq \mathbf{H}_{/X}(f^* U \times A, B) \\ & \stackrel{}{\to} \mathbf{H}_{/Y}( \sum_f( f^* U \times A ), \sum_f B ) \\ & \stackrel{Frob.Rec.}{\simeq} \mathbf{H}_{/Y}( U \times \sum_f A , \sum_f B ) \\ & \simeq \mathbf{H}_{/Y}(U, \sum_f A , \sum_f B) \end{aligned}

where the first and the last steps use adjunction properties, where the morphism in the middle is the component of the dependent sum functor, and where “Frob.Rec.” is Frobenius reciprocity.

Since this is natural in $U$, the Yoneda lemma implies the claimed morphism.

###### Proof/Construction 2

There is the composite morphism

$\left(f^\ast \prod_f [A, B]\right) \times A \stackrel{counit \times id_A}{\to} [A, B] \times A \stackrel{eval}{\to} B \stackrel{unit}{\to} f^\ast \sum_f B$

of the adjunction (co)units and the evaluation map of the internal hom. Its hom-adjunct is

$A \to [f^\ast \prod_f [A, B], f^\ast \sum_f B] \cong f^\ast [\prod_f [A, B], \sum_f B] \,,$

using prop. 5 on the right. The hom-adjunct of that in turn is

$\sum_f A \to [\prod_f [A, B], \sum_f B]$

and by symmetry the morphism that we are after:

$\prod_f [A, B] \to [\sum_f A, \sum_f B] \,.$
###### Remark

If $Y$ is the terminal object (for simplicity), then the morphism of prop. 6 can be understood as follows: a global element of the dependent product $\prod_f [A,B]$ is given by a commuting diagram in $\mathbf{H}$ of the form

$\array{ \sum_f A &&\to&& \sum_f B \\ & \searrow && \swarrow \\ && X } \,.$

The map in prop. 6 picks out the top horizontal morphism in this diagram.

### For smooth spaces and smooth $\infty$-groupoids

Consider the site $C =$ SmthMfd of smooth manifolds (and the open cover coverage) or equivalently over the dense subsite CartSp of Cartesian spaces and smooth functions between these.

The sheaf topos/(∞,1)-sheaf (∞,1)-topos $\mathbf{H} = Sh(C)$ is that of smooth spaces/smooth ∞-groupoids. So the discussion of internal homs here is a special case of the above discussion In a sheaf topos.

###### Example

For $X , Y \in SmthMfd \hookrightarrow \mathbf{H}$ two smooth manifolds, the internal hom $[X,Y] \in \mathbf{H}$ is the mapping space between them regarded as a diffeological space.

See at manifold structure of mapping spaces for when this internal hom is representable again by a smooth manifold.

###### Example

For $X \in SmthMfd \hookrightarrow \mathbf{H}$ the internal automorphism group, example 1, of $X$ is the diffeomorphism group of $X$, regarded as a diffeological group

$\mathbf{Aut}(X) = \mathbf{Diff}(X) \,.$

### For super vector spaces

The category $SVect$ of super vector spaces is the category of $\mathbb{Z}/2$-graded vector spaces. Thus, its objects are pairs of vector spaces $(V_+,V_-)$, with $V_+$ called the even part and $V_-$ the odd part. The morphisms in $SVect$ are likewise pairs of linear maps, i.e. we define $SVect$ to be $Vect \times Vect = Vect^{\mathbb{Z}/2}$, as usual for any sort of graded object. With this definition of the category $SVect$, we capture the concepts of superalgebra and so on in succinct categorical terms.

Because the morphisms in $SVect$ send even things to even things and odd things to odd things, they are sometimes called even linear maps, and one may write

$SVect(V, W) = Even Lin(V,W).$

Note that $SVect$ is enriched over $Vect$, i.e. these hom-sets are vector spaces.

Occasionally, however, one does need to refer to the odd linear maps, which send even things to odd things and odd things to even things. That is, an odd linear map $V\to W$ is a pair of linear maps $V_+ \to W_-$ and $V_-\to W_+$. The internal-hom in $SVect$ allows us to capture these as well: it is the following super vector space:

$[V,W]_+ = Even Lin(V,W) \qquad [V,W]_- = Odd Lin(V,W).$

With this definition, $SVect$ becomes a closed monoidal category.

We can equivalently regard a super vector spaces $(V_+,V_-)$ as being the direct sum vector space $V_+ \oplus V_-$ equipped with this direct sum decomposition. If we view the internal-hom $[V,W]$ in this way as well, then we have

$[V, W] = Even Lin(V,W) \oplus Odd Lin(V,W) = Lin(V,W).$

In other words, any linear map between these “summed” super vector spaces decomposes uniquely as the sum of an even linear map and an odd one.

### For Banach spaces

A similar thing happens in the category $Ban$ of Banach spaces and short linear operators. The external hom consists of only the short linear maps (those bounded by $1$):

$Ban(V,W) = \{ f\colon Lin(V,W) \;|\; {\|f\|} \leq 1 \} .$

This definition of morphism recovers the most specific notion of isomorphism of Banach spaces, as well as defining the product and coproduct as the direct sum completed with $p = \infty$ or $p = 1$ respectively.

But the internal hom is the Banach space of all bounded linear maps:

$[V,W] = \{ f\colon Lin(V,W) \;|\; {\|f\|} \lt \infty \} .$

This is a Banach space and makes $Ban$ into a closed category.

## References

A discussion query (R. Brown, T. Bartels, M. Shulman) about internal hom is at $n$Forum here.

Revised on April 17, 2015 16:00:22 by Urs Schreiber (195.113.30.252)