An infinitary Lawvere theory is a generalisation of a Lawvere theory to allow infinitary operations. As a motivating example, consider the theory of suplattices, where we have, for every cardinal number , an operation to take the supremum of elements.
Size issues can be tricky for infinitary Lawvere theories. Naïvely, you would expect an infinitary Lawvere theory of complete lattices too, but there is none, because a smallness condition fails. (This is connected to the nonexistence of free complete lattices.)
In accordance with the red herring principle, an infinitary Lawvere theory should not be thought of a special case of a Lawvere theory. To avoid this, one may call the latter a finitary Lawvere theory.
There are also many-sorted infinitary Lawvere theories, as well as -ary Lawvere theories for any regular cardinal .
A many-sorted infinitary Lawvere theory is a locally small category with all small products and with a small collection of objects (called sorts) such that every object is a small product of these sorts.
Given a small set , an -sorted infinitary Lawvere theory is a locally small category with all small products and equipped with an -indexed family of objects (called sorts) such that every object is a small product of these sorts. (Note that an -indexed family of objects is precisely a functor to from the discrete category on .)
If is an -sorted infinitary Lawvere theory, then is a many-sorted infinitary Lawvere theory; conversely, any many-sorted infinitary Lawvere theory may be interpreted as an -sorted infinitary Lawvere theory, where is (the set of isomoprhism classes of) an appropriate family and the -indexed family is given by the identity indexing.
An unsorted infinitary Lawvere theory is a locally small category with all small products and equipped with an object (so that is a pointed category) such that every object is isomorphic to for some small cardinal number . An infinitary Lawvere theory is by default an unsorted infinitary Lawvere theory, invoking the red herring principle. Note that an unsorted infinitary Lawvery theory is the same thing as a -sorted infinitary Lawvere theory.
We will give the definitions of further special cases for unsorted infinitary Lawvere theories, but these definitions may also be generalised to the many-sorted case.
Given a regular cardinal , a -ary Lawvere theory is a locally small pointed category with all -ary products for such that every object is isomorphic to for some . Every -ary Lawvere theory may be interpreted as an infinitary Lawvere theory by using the free category with all small products on . More generally, every -ary Lawvere theory may be interpreted as a -ary Lawvere theory if .
A bounded infinitary Lawvere theory is an infinitary Lawvere theory which is (under the interpretation above) equivalent to some -ary Lawvere theory.
A finitary Lawvere theory is a locally small pointed category with all finitary products such that every object is isomorphic to for some natural number . A Lawvere theory is by default a finitary Lawvere theory, invoking the red herring principle. Note that a finitary Lawvere theory is the same thing as an -ary Lawvere theory.
Freyd and Street (1995) have shown that a category is small if and only if both and the functor category are locally small. Analogously, it seems that a category with products may be given the structure of a many-sorted infinitary Lawvere theory if and only if both and (the category of product-preserving functors from to , a full subcategory of ) are locally small. In both cases, the ‘only if’ part is straightfoward, but we haven’t yet proved the ‘if’ part.
See the n-Forum for more preliminary results.
Let us write this functor category as and start by showing that it is a locally small category.
The category of product-preserving functors and natural transformations is locally small.
Let be two product-preserving covariant functors. From the definition of a multi-sorted infinitary category we see that we may choose a set of sorts, say . Let be the image of .
As is a set, the product
is a set. We define a function by sending a natural transformation to the product of its values at each . That is,
Let be an arbitrary object. From the definition of a multi-sorted infinitary Lawvere theory, there is an isomorphism
for some sets . For each and , there is a projection . As and are product-preserving, we have isomorphisms
commuting with the projections. Thus for each and , we have the following commutative diagram.
Since this holds for all and , by the basic properties of products, there is a unique morphism making the diagram commute. This morphism is normally written . Thus under the isomorphism above, is taken to . As this holds for all , is completely determined by the , whence the map in (1) is injective.
Hence is a subset of a set and thus a set.
Now that we have a locally small category, the next step is to show that the forgetful functor has a left adjoint. Firstly, we need to define this forgetful functor. As we are in the most general case, the forgetful functor does not go to but to -indexed tuples of sets (or an -graded set), where is a choice of set of sorts for . To make things a little simpler, we assume that the sorting functor is injective on isomorphism classes so that if in then in . With this assumption, the forgetful functor is the evaluation functor:
The forgetful functor has a left adjoint. The left adjoint is given on objects by:
To ease the notation, let us write for the forgetful (“underlying set(s)”) functor and for the free functor.
To show the adjunction, we define the unit and counit natural transformations:
Let us consider . To define this, we evaluate it at an -graded set, , to get a morphism of graded sets, . Such a morphism is itself a natural transformation so we evaluate again at . At this point, we have a function (in )
This is simple to describe: it is assignment to of the projection onto the th -factor:
The counit, , is a little more complicated to describe. Let be a covariant product-preserving functor. Evaluated at , is a natural transformation
By the Yoneda lemma, such natural transformations are in bijective correspondence with elements of . As is product-preserving, there is a natural isomorphism:
Now for any set , there is an obvious element in : the one that has a in the th place (which corresponds to the identity function ). Taking the product of these gives an element in which we use to define .
Let us now prove that these provide the desired adjunction. For one half, we need to consider the composition:
Let us apply this to a graded set . Then we are looking at a natural transformation from
to itself. By the standard Yoneda argument, it is sufficient to look at the induced function on
and in particular at the image of the identity therein.
The first part comes from at . For each , we have a function and thus a morphism
What this does is the following: it sends component corresponding to the th projection to the th component and all other components are forgotten. Thus we have a morphism:
Under this, the identity morphism goes to the projection morphism described just above.
Now we apply to get a morphism:
To find this element, we consider the diagram:
In this diagram, we have left off the subscript on for conciseness. The vertical morphism is that induced by the projection from (2). Since we want to have this projection itself in the lower-left, an obvious place to start is with the identity in the top-left. The right-hand square commutes since the vertical maps are projections. Starting with the identity in the top-left, we get the “Yoneda element” corresponding to in the top-right. That element can be written . The vertical map selects the th element of the list and puts it in the th slot. As this is the projection , when moving back to the lower-middle, we obtain the identity morphism as required.
Now let us turn to the other half. We need to consider the composition:
So we need to start with a covariant product-preserving functor and apply . Then we are looking at a morphism of graded sets from
to itself. So we pick and look at the function
We start with the function defined by the unit :
which sends to the projection which corresponds to the th factor of the th factor.
Now we apply . This gives us a natural transformation
which we evaluate at (technically, which we evaluate at which we then evaluate at ). This natural transformation is determined by an element of and the element that we want is the image of that element under the function induced by the projection:
As is product-preserving, we can rearrange this to:
The projection selects the th term of the element in question, and this element is, by definition, . Hence the induced function on is the identity and the adjunction is shown.