Stable homotopy theory
For a category and two objects, the internal hom from to is, if it exists, another object of which behaves like the “object of morphisms” from to . In other words it is, if it exists, an internal version of the ordinary hom set or more generally hom object of a locally small category or -enriched category.
One way to make this precise is to mimic the basic property of a function set of functions between two sets and : that is uniquely characterizes by the fact that for any other set the functions are in natural bijection with the functions out of the cartesian product of with . Formally this says that the functor of taking the cartesian product with the set has a right adjoint given by the construction .
This, then, is, generally, the definition of internal hom in any cartesian monoidal category or in fact in any monoidal category : the right adjoint to the given tensor product functor for all objects . It may or may not exist. If it exists, one says that is a closed monoidal category. Explicity, the condition is that there is an isomorphism(bijection)
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 -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 in that
using that 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.
Let be a monoidal category. An internal hom in is a functor
such that for every object we have a pair of adjoint functors
If this exists, is called a closed monoidal category.
Let be a closed monoidal category.
For two objects, the evaluation map
is the -adjunct of the identity .
Let be a closed monoidal category.
For three objects, the composition morphism
is the -adjunct of the following composite of two evaluation maps, def. 2:
Relation to function types
The internal hom is the categorical semantics of what in type theory are function types
Induced monad (state monad)
For each object the (internal hom tensor product)-adjunction induces a monad . In computer science this monad (in computer science) is called the state monad.
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 -sheaf -topos
Let be a site. Let be the sheaf topos over 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.
Hence the internal hom exist.
For two objects, the internal hom-object
is the sheaf/(∞,1)-sheaf given by the assignment
for all objects which on the right we regard under the Yoneda embedding/∞-Yoneda embedding .
See also at closed monoidal structure on presheaves.
By the Yoneda lemma/(∞,1)-Yoneda lemma we have natural equivalences
and by the defining adjunction this is naturally equivalent to
For , the evaluation map, def. 2,
is the morphism of sheaves which over each sends a morphism of sheaves (which is the first component by prop. 2) and an to
See (MacLane-Moerdijk, p. 46).
For three objects of , the canonical composition morphism, def. 3,
is given by the morphism of presheaves/(∞,1)-presheaves whose component over is the morphism of sets/∞-groupoids
which sends a pair to the composite
where is the diagonal morphism on .
By definition 3 the morphism is the adjunct of the double evaluation map
Since the cartesian product of two sheaves is computed objectwise
it follows that over each this double evaluation map is the morphism of sets/-groupoids
hence by prop. 2
where now by prop. \ref this is the external evaluation.
In slice categories
Let be a locally cartesian closed category. This means that for each object the slice category is a cartesian closed category. The product in the slice is given by the fiber product over computed in . Fairly detailed discussion of constructions of the internal hom in such slices is at locally cartesian closed category – cartesian closure in terms of base change and dependent product.
We record some further properties
This is discussed in more detail at cartesian closed functor – Examples.
So for we have isomorphisms
between the image of the internal hom under and the internal hom of the images of and separately.
For a locally cartesian closed category, any morphism, and two objects in the slice over , there is a natural morphism
Here are two ways to get this morphism:
For any object we have a canonical morphism of hom sets
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 , the Yoneda lemma implies the claimed morphism.
There is the composite morphism
of the adjunction (co)units and the evaluation map of the internal hom. Its hom-adjunct is
using prop. 5 on the right. The hom-adjunct of that in turn is
and by symmetry the morphism that we are after:
For smooth spaces and smooth -groupoids
Consider the site 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 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.
For the internal automorphism group, example 1, of is the diffeomorphism group of , regarded as a diffeological group
For chain complexes
For super vector spaces
The category of super vector spaces is the category of -graded vector spaces. Thus, its objects are pairs of vector spaces , with called the even part and the odd part. The morphisms in are likewise pairs of linear maps, i.e. we define to be , as usual for any sort of graded object. With this definition of the category , we capture the concepts of superalgebra and so on in succinct categorical terms.
Because the morphisms in send even things to even things and odd things to odd things, they are sometimes called even linear maps, and one may write
Note that is enriched over , 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 is a pair of linear maps and . The internal-hom in allows us to capture these as well: it is the following super vector space:
With this definition, becomes a closed monoidal category.
We can equivalently regard a super vector spaces as being the direct sum vector space equipped with this direct sum decomposition. If we view the internal-hom in this way as well, then we have
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 of Banach spaces and short linear operators. The external hom consists of only the short linear maps (those bounded by ):
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 or respectively.
But the internal hom is the Banach space of all bounded linear maps:
This is a Banach space and makes into a closed category.
A discussion query (R. Brown, T. Bartels, M. Shulman) about internal hom is at Forum here.