# nLab closed monoidal structure on presheaves

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

category theory

# Contents

## Idea

As every topos, the category $\mathrm{PSh}\left(X\right)$ of presheaves is cartesian closed monoidal.

## Definition

Let $S$ be a category.

The standard monoidal structure on presheaves $\mathrm{PSh}:=\left[{S}^{\mathrm{op}},\mathrm{Set}\right]$ is the cartesian monoidal structure.

Recalling that limits of presheaves are computed objectwise, this is the pointwise cartesian product in Set: for two presheaves $F,G$ their product presheaf $F×G$ is given by

$F×G:U↦F\left(U\right)×G\left(U\right)\phantom{\rule{thinmathspace}{0ex}},$F \times G : U \mapsto F(U) \times G(U) \,,

where on the right the product is in Set.

###### Proposition

The corresponding internal hom

$\left[-,-\right]:{\mathrm{PSh}}^{\mathrm{op}}×\mathrm{PSh}\to \mathrm{PSh}$[-,-] : PSh^{op} \times PSh \to PSh

exists and is given by

$\left[F,G\right]:U↦{\mathrm{Hom}}_{\mathrm{PSh}}\left(Y\left(U\right)×F,G\right)\phantom{\rule{thinmathspace}{0ex}},$[F,G] : U \mapsto Hom_{PSh}( Y(U)\times F, G ) \,,

where $Y:S\to \left[{S}^{\mathrm{op}},\mathrm{Set}\right]$ is the Yoneda embedding.

###### Proof

First assume that $\left[F,G\right]$ exists, so that by the hom-adjunction isomorphism we have $\mathrm{Hom}\left(R,\left[F,G\right]\right)\simeq \mathrm{Hom}\left(R×F,G\right)$. In particular, for each representable functor $R=Y\left(U\right)$ (with $Y$ the Yoneda embedding) and using the Yoneda lemma we get

$\begin{array}{rl}\left[F,G\right]\left(U\right)& \simeq \mathrm{Hom}\left(Y\left(U\right),\left[F,G\right]\right)\\ & \simeq \mathrm{Hom}\left(Y\left(U\right)×F,G\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} [F,G](U) & \simeq Hom(Y(U), [F,G]) \\ & \simeq Hom(Y(U) \times F, G) \end{aligned} \,.

So if the internal hom exists, it has to be of the form given. It remains to show that with this definition $\left[F,-\right]$ really is right adjoint to $-\otimes F$.

## Definition in terms of homs of direct images

Often another, equivalent, expression is used to express the internal hom of presheaves:

Let $X$ be a pre-site with underlying category ${S}_{X}$. Recall from the discussion at site that just means that we have a category ${S}_{X}$ on which we consider presheaves $F\in \mathrm{PSh}\left({S}_{X}\right):=\left[{S}_{X}^{\mathrm{op}},\mathrm{Set}\right]$, but that it suggests that

• to each object $U\in \mathrm{PSh}\left(X\right)$ and in particular to each $U\in {S}_{X}↪\mathrm{PSh}\left(X\right)$ there is naturally associated the pre-site $U$ with underlying category the comma category ${S}_{U}=\left(Y/Y\left(U\right)\right)$;

• that the canonical forgetful functor ${j}_{U\to X}^{t}:{S}_{U}\to {S}_{X}$, which can be thought of as a morphism of pre-sites ${j}_{U\to X}:X\to U$ induces the direct image functor $\left({j}_{U\to X}{\right)}_{*}:\mathrm{PSh}\left(X\right)\to \mathrm{PSh}\left(U\right)$ which we write $F↦F{\mid }_{U}$.

Then in these terms the above internal hom for presheaves

$\mathrm{hom}:\mathrm{PSh}\left(X{\right)}^{\mathrm{op}}×\mathrm{PSh}\left(X\right)\to \mathrm{PSh}\left(X\right)$hom : PSh(X)^{op} \times PSh(X) \to PSh(X)

is expressed for all $F,G\in \mathrm{PSh}\left(X\right)$ by

$\mathrm{hom}\left(F,G\right):U↦{\mathrm{Hom}}_{\mathrm{PSh}\left(U\right)}\left(F{\mid }_{U},G{\mid }_{U}\right)\phantom{\rule{thinmathspace}{0ex}}.$hom(F,G) : U \mapsto Hom_{PSh(U)}(F|_U, G|_U) \,.

## Relation of the two definitions

To see the equivalence of the two definitions, notice

• that by the Yoneda lemma we have that ${S}_{U}$ is simply the over category ${S}_{U}={S}_{X}/U$;
• by the general properties of functors and comma categories there is an equivalence $\mathrm{PSh}\left({S}_{X}/U\right)\simeq \mathrm{PSh}\left({S}_{X}\right)/y\left(U\right)$;
• which identifies the functor $\left(-\right){\mid }_{U}:\mathrm{PSh}\left({S}_{X}\right)\to \mathrm{PSh}\left({S}_{U}\right)$ with the functor $\left(\left(-\right)×y\left(U\right)\stackrel{{p}_{2}}{\to }y\left(U\right)\right):\mathrm{PSh}\left({S}_{X}\right)\to \mathrm{PSh}\left({S}_{X}\right)/y\left(U\right)$;
• and that ${\mathrm{Hom}}_{\mathrm{PSh}\left({S}_{X}\right)/y\left(U\right)}\left(y\left(U\right)×F,y\left(U\right)×G\right)\simeq {\mathrm{Hom}}_{\mathrm{PSh}\left({S}_{X}\right)}\left(y\left(U\right)×F,G\right)$.

## References

The first definition is discussed for instance in section I.6 of

The second definition is discussed for instance in section 17.1 of

Revised on September 26, 2012 03:01:57 by Urs Schreiber (82.169.65.155)