This is an introductory post about some easy examples of internal Hom.
Definition. Let
be a symmetric monoidal category, i.e. a category
with a functor
that is associative, unital, and commutative up to natural isomorphism. Then an internal Hom in
is a functor
![Rendered by QuickLaTeX.com \[\mathbf{Hom}(-,-) \colon \mathscr C\op \times \mathscr C \to \mathscr C\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-c14c9c14c53444e8e11893f5016cfc5a_l3.svg)
such that
is a left adjoint to
for any
, i.e. there are functorial isomorphisms
![Rendered by QuickLaTeX.com \[\operatorname{Hom}(X \otimes Y, Z) \stackrel\sim\to \operatorname{Hom}(X, \mathbf{Hom}(Y,Z)).\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-a9e9f03ab6dd48790300b0bb2a587c9a_l3.svg)
Remark. In the easiest examples, we typically think of
as ‘upgrading
to an object of
‘:
Example. Let
be a commutative ring, and let
be the category of
-modules, with
the tensor product. Then
with its natural
-module structure is an internal Hom, by the usual tensor-Hom adjunction:
![Rendered by QuickLaTeX.com \[\operatorname{Hom}_R(M \otimes_R N, K) \cong \operatorname{Hom}_R(M, \mathbf{Hom}(N, K)).\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-c053d9b00b75a3814c80d88d9e6e646b_l3.svg)
The same is true when
is the category of
-bimodules for a not necessarily commutative ring
.
However, we cannot do this for left
-modules over a noncommutative ring, because there is no natural
-module structure on
for left
-modules
and
. In general, the tensor product takes an
-bimodule
and a
-bimodule
and produces an
-bimodule
. Taking
gives a way to tensor a right
-module with a left
-module, but there is no standard way to tensor two left
-modules, let alone equip it with the structure of a left
-module.
Example. Let
. Then
is naturally a set, making it into an internal Hom for
:
![Rendered by QuickLaTeX.com \[\operatorname{Hom}(X \times Y, Z) \stackrel\sim\to \operatorname{Hom}(X, \mathbf{Hom}(Y,Z)).\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-724d0507fa56e05fa53fbcc4a2af891d_l3.svg)
When
is the categorical product
, the internal
(if it exists) is usually called an exponential object, in analogy with the case
above.
Example. Another example of exponential objects is from topology. Let
be the category of locally compact Hausdorff topological spaces. Then the compact-open topology makes
into an internal Hom of topological spaces. (There are mild generalisations of this beyond the compact Hausdorff case, but for an arbitrary topological space
the functor
does not preserve colimits and hence cannot admit a right adjoint.)
Example. An example of a slightly different nature is chain complexes: let
be a commutative ring, and let
be the category of cochain complexes
![Rendered by QuickLaTeX.com \[\ldots \to C^{i-1} \to C^i \to C^{i+1} \to \ldots\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-b11be861c86069e855982141f08aae85_l3.svg)
of
-modules (meaning each
is an
-module, and the
are
-linear maps satisfying
). Homomorphisms
are commutative diagrams
![Rendered by QuickLaTeX.com \[\begin{array}{ccccccc}\ldots & \to & C^i & \to & C^{i+1} & \to & \ldots \\ & & \!\!\!\!\! f^i\downarrow & & \downarrow f^{i+1}\!\!\!\!\!\!\! & & \\ \ldots & \to & D^i & \to & D^{i+1} & \to & \ldots,\!\!\end{array}\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-82b0b03f9e235983432f4ceb02a11492_l3.svg)
and the tensor product is given by the direct sum totalisation of the double complex of componentwise tensor products.
There isn’t a natural way to ‘endow
with the structure of a chain complex’, but there is an internal Hom given by
![Rendered by QuickLaTeX.com \[\mathbf{Hom}(C, D)^i = \prod_{m \in \mathbf Z} \operatorname{Hom}(C_m, D_{m+i}),\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-3cf574776217419a8383f65181d32a09_l3.svg)
with differentials given by
![Rendered by QuickLaTeX.com \[d^if = d_D f - (-1)^i f d_C.\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-4c553e282de32891abf5bf1c6139c14c_l3.svg)
Then we get for example
![Rendered by QuickLaTeX.com \[\operatorname{Hom}(R[0], \mathbf{Hom}(C, D)) \cong \operatorname{Hom}(C, D),\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-ced2185b01ff3da3f98eb6a02ef28dc1_l3.svg)
since a morphism
is given by an element
such that
, i.e.
, meaning that
is a morphism of cochain complexes.
Example. The final example for today is presheaves and sheaves. If
is a topological space, then the category
of abelian sheaves on
has an internal Hom given by
![Rendered by QuickLaTeX.com \[\mathbf{Hom}(\mathscr F, \mathscr G)(U) = \operatorname{Hom}(\mathscr F|_U, \mathscr G|_U),\]](https://lovelylittlelemmas.rjprojects.net/wp-content/ql-cache/quicklatex.com-9a82660bf3604fb7765d2d8b4e78a5cc_l3.svg)
with the obvious transition maps for inclusions
of open sets. This is usually called the sheaf Hom. A similar statement holds for presheaves.