infinitary Lawvere theory

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 $n$, an operation to take the supremum of $n$ elements.

Size issues can be tricky for infinitary Lawvere theories. NaΓ―vely, 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 $\kappa$-ary Lawvere theories for any regular cardinal $\kappa$.

A **many-sorted infinitary Lawvere theory** is a locally small category $\mathcal{D}$ with all small products and with a small collection $\mathcal{R}$ of objects (called **sorts**) such that every object is a small product of these sorts.

Given a small set $S$, an **$S$-sorted infinitary Lawvere theory** is a locally small category $\mathcal{D}$ with all small products and equipped with an $S$-indexed family $\mathcal{R}$ of objects (called **sorts**) such that every object is a small product of these sorts. (Note that an $S$-indexed family of objects is precisely a functor to $\mathcal{D}$ from the discrete category on $S$.)

If $(\mathcal{D},\mathcal{R})$ is an $S$-sorted infinitary Lawvere theory, then $\mathcal{D}$ is a many-sorted infinitary Lawvere theory; conversely, any many-sorted infinitary Lawvere theory may be interpreted as an $S$-sorted infinitary Lawvere theory, where $S$ is (the set of isomoprhism classes of) an appropriate family $\mathcal{R}$ and the $S$-indexed family is given by the identity indexing.

An **unsorted infinitary Lawvere theory** is a locally small category $\mathcal{D}$ with all small products and equipped with an object $R$ (so that $(\mathcal{D},R)$ is a pointed category) such that every object is isomorphic to $R^n$ for some small cardinal number $n$. 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 $1$-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 $\kappa$, a **$\kappa$-ary Lawvere theory** is a locally small pointed category $(\mathcal{D},R)$ with all $n$-ary products for $n \lt \kappa$ such that every object is isomorphic to $R^n$ for some $n \lt \kappa$. Every $\kappa$-ary Lawvere theory may be interpreted as an infinitary Lawvere theory by using the free category with all small products on $\mathcal{D}$. More generally, every $\kappa$-ary Lawvere theory may be interpreted as a $\lambda$-ary Lawvere theory if $\kappa \leq \lambda$.

A **bounded infinitary Lawvere theory** is an infinitary Lawvere theory which is (under the interpretation above) equivalent to some $\kappa$-ary Lawvere theory.

A **finitary Lawvere theory** is a locally small pointed category $(\mathcal{D},R)$ with all finitary products such that every object is isomorphic to $R^n$ for some natural number $n$. 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 $\aleph_0$-ary Lawvere theory.

Freyd and Street (1995) have shown that a category $\mathcal{D}$ is small if and only if both $\mathcal{D}$ and the functor category $[\mathcal{D},Set]$ are locally small. Analogously, it seems that a category $\mathcal{D}$ with products may be given the structure of a many-sorted infinitary Lawvere theory if and only if both $\mathcal{D}$ and $Prod[\mathcal{D},Set]$ (the category of product-preserving functors from $\mathcal{D}$ to $Set$, a full subcategory of $[\mathcal{D},Set]$) are locally small. In both cases, the βonly ifβ part is straightfoward, but we havenβt yet proved the βifβ part.

See the n-Forum for more preliminary results.

A multi-sorted infinitary Lawvere theory $\mathcal{D}$ defines a monadic category over Set by taking the functor category consisting of all **product-preserving** covariant functors from $\mathcal{D}$ in to $Set$.

Let us write this functor category as $Prod[\mathcal{D},Set]$ and start by showing that it is a locally small category.

The category $Prod[\mathcal{D},Set]$ of product-preserving functors and natural transformations is locally small.

Let $V_1, V_2 \colon \mathcal{D} \to Set$ 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 $S$. Let $D_s \in \mathcal{D}$ be the image of $s \in S$.

As $S$ is a set, the product

$\prod_{s \in S} Set(V_1(D_s), V_2(D_s))$

is a set. We define a function $[V_1,V_2] \to \prod_{s \in S} Set(V_1(D_s), V_2(D_s))$ by sending a natural transformation $\alpha \colon V_1 \to V_2$ to the product of its values at each $D_s$. That is,

(1)$\alpha \mapsto \big(\alpha_{D_s}\big)_{s \in S}.$

Let $D \in \mathcal{D}$ be an arbitrary object. From the definition of a multi-sorted infinitary Lawvere theory, there is an isomorphism

$d \colon D \cong \prod_{s \in S} D_s^{X_s}$

for some sets $X_s$. For each $s \in S$ and $x \in X_s$, there is a projection $p_{s,x} \colon \prod_{s \in S} D_s^{X_s} \to D_s$. As $V_1$ and $V_2$ are product-preserving, we have isomorphisms

$\cong \prod_{s \in S} V_i(D_s)^{X_s}$

commuting with the projections. Thus for each $s \in S$ and $x \in X_s$, we have the following commutative diagram.

Since this holds for all $s \in S$ and $x \in X_s$, by the basic properties of products, there is a unique morphism $\prod_{s \in S} V_1(D_s)^{X_s} \to \prod_{s \in S} V_2(D_s)^{X_s}$ making the diagram commute. This morphism is normally written $\prod_{s \in S} \alpha_{D_s}^{X_s}$. Thus under the isomorphism $d$ above, $\alpha_D$ is taken to $\prod_{s \in S} \alpha_{D_s}^{X_s}$. As this holds for all $D \in \mathcal{D}$, $\alpha$ is completely determined by the $\alpha_{D_s}$, whence the map in (1) is injective.

Hence $[V_1,V_2]$ 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 $Set$ but to $S$-indexed tuples of sets (or an $S$-graded set), where $S$ is a choice of set of sorts for $\mathcal{D}$. To make things a little simpler, we assume that the *sorting* functor $S \to \mathcal{D}$ is injective on isomorphism classes so that if $s \ne s'$ in $S$ then $D_s \ncong D_{s'}$ in $\mathcal{D}$. With this assumption, the forgetful functor $Prod[\mathcal{D},Set] \to Set^S$ is the evaluation functor:

$V \mapsto \big(s \mapsto V(D_s)\big), \qquad \alpha \mapsto \big(s \mapsto \alpha_{D_s}\big).$

The forgetful functor $Prod[\mathcal{D},Set] \to Set^S$ has a left adjoint. The left adjoint is given on objects by:

$\big(s \mapsto X_s\big) \mapsto \mathcal{D}\left(\prod_{s \in S} D_s^{X_s}, -\right)$

To ease the notation, let us write $U$ for the forgetful (βunderlying set(s)β) functor and $F$ for the free functor.

To show the adjunction, we define the unit and counit natural transformations:

$\eta \colon I \to U F, \qquad \epsilon \colon F U \to I$

Let us consider $\eta$. To define this, we evaluate it at an $S$-graded set, $X$, to get a morphism of graded sets, $\eta_X$. Such a morphism is itself a natural transformation so we evaluate again at $s_0 \in S$. At this point, we have a function (in $Set$)

$\eta_{X,s_0} \colon X_{s_0} \to \mathcal{D}\left(\prod_{s \in S} D_s^{X_s}, D_{s_0}\right).$

This is simple to describe: it is assignment to $x \in X_{s_0}$ of the projection onto the $x$th $D_{s_0}$-factor:

$\prod_{s \in S} D_s^{X_s} \to D_{s_0}^{X_{s_0}} \overset{p_x}{\to} D_{s_0}.$

The counit, $\epsilon$, is a little more complicated to describe. Let $V \colon \mathcal{D} \to Set$ be a covariant product-preserving functor. Evaluated at $V$, $\epsilon_V$ is a natural transformation

$\mathcal{D}\left(\prod_{s \in S} D_s^{V(D_s)}, -\right) \to V$

By the Yoneda lemma, such natural transformations are in bijective correspondence with elements of $V\left(\prod_{s \in S} D_s^{V(D_s)}\right)$. As $V$ is product-preserving, there is a natural isomorphism:

$V\left(\prod_{s \in S} D_s^{V(D_s)}\right) \cong \prod_{s \in S} V(D_s)^{V(D_s)}$

Now for any set $Z$, there is an obvious element in $Z^Z$: the one that has a $z$ in the $z$th place (which corresponds to the identity function $Z \to Z$). Taking the product of these gives an element in $\prod_{s \in S} V(D_s)^{V(D_s)}$ which we use to define $\epsilon_V$.

Let us now prove that these provide the desired adjunction. For one half, we need to consider the composition:

$F \overset{F \eta}{\to} F U F \overset{\epsilon F}{\to} F$

Let us apply this to a graded set $X = (s \mapsto X_s)$. Then we are looking at a natural transformation from

$\mathcal{D}\left(\prod_{s \in S} D_s^{X_s}, -\right)$

to itself. By the standard Yoneda argument, it is sufficient to look at the induced function on

$\mathcal{D}\left(\prod_{s \in S} D_s^{X_s}, \prod_{s \in S} D_s^{X_s}\right)$

and in particular at the image of the identity therein.

The first part comes from $F\eta$ at $F(X)$. For each $s_0 \in S$, we have a function $\eta_{X,s_0} \colon X_{s_0} \to \mathcal{D}\left(\prod_{s \in S} D_s^{X_s}, D_{s_0}\right)$ and thus a morphism

(2)$\prod_{s \in S} D_s^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)} \to \prod_{s \in S} D_s^{X_s}$

What this does is the following: it sends component corresponding to the $(s,x)$th projection to the $x$th component and all other components are forgotten. Thus we have a morphism:

$\mathcal{D}\left( \prod_{s \in S} D_s^{X_s}, \prod_{s \in S} D_s^{X_s}\right) \to \mathcal{D}\left(
\prod_{s \in S} D_s^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)}, \prod_{s \in S} D_s^{X_s}\right)$

Under this, the identity morphism goes to the projection morphism described just above.

Now we apply $\epsilon F$ to get a morphism:

$\mathcal{D}\left(
\prod_{s \in S} D_s^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)}, \prod_{s \in S} D_s^{X_s}\right)
\to \mathcal{D}\left( \prod_{s \in S} D_s^{X_s}, \prod_{s \in S} D_s^{X_s}\right)$

To find this element, we consider the diagram:

$\array{
\mathcal{D}\left(
\prod_{s \in S} D_s^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)},
\prod_{s \in S} D_s^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)}\right)
&\overset{\epsilon }{\to}&
\mathcal{D}\left( \prod_{s \in S} D_s^{X_s},
\prod_{s \in S} D_s^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)}\right)
& \cong &
\prod_{s \in S} \mathcal{D}\left( \prod_{s'' \in S} D_{s''}^{X_{s''}},
D_s\right)^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)} \\
\downarrow & & \downarrow & & \downarrow \\
\mathcal{D}\left(
\prod_{s \in S} D_s^{\mathcal{D}\left(\prod_{s' \in S} D_{s'}^{X_{s'}},D_s\right)}, \prod_{s \in S} D_s^{X_s}\right)
&\overset{\epsilon }{\to}&
\mathcal{D}\left( \prod_{s \in S} D_s^{X_s}, \prod_{s \in S} D_s^{X_s}\right)
& \cong &
\prod_{s \in S}\mathcal{D}\left( \prod_{s'' \in S} D_{s''}^{X_{s''}}, D_s\right)^{X_s}
}$

In this diagram, we have left off the subscript on $\epsilon$ 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 βYoneda elementβ corresponding to $\epsilon$ in the top-right. That element can be written $(f)_f$. The vertical map selects the $p_{s,x}$th element of the list and puts it in the $(s,x)$th slot. As this is the projection $p_{s,x}$, 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:

$U \overset{\eta U}{\to} U F U \overset{U \epsilon}{\to} U$

So we need to start with a covariant product-preserving functor $V \colon \mathcal{D} \to Set$ and apply $U$. Then we are looking at a morphism of graded sets from

$\big(s \mapsto V(D_s)\big)$

to itself. So we pick $s_0 \in S$ and look at the function

$V(D_{s_0}) \to \mathcal{D}\left(\prod_{s \in S} D_s^{V(D_s)}, D_{s_0}\right) \to V(D_{s_0}).$

We start with the function defined by the unit $\eta$:

$V(D_{s_0}) \to \mathcal{D}\left(\prod_{s \in S} D_s^{V(D_s)}, D_{s_0}\right)$

which sends $v \in V(D_{s_0})$ to the projection $\prod_{s \in S} D_s^{V(D_s)} \to D_{s_0}$ which corresponds to the $v$th factor of the $s_0$th factor.

Now we apply $\epsilon$. This gives us a natural transformation

$\mathcal{D}\left(\prod_{s \in S} D_s^{V(D_s)}, - \right) \to V(-)$

which we evaluate at $D_{s_0}$ (technically, which we evaluate at $(s \mapsto D_s)$ which we then evaluate at $s_0$). This natural transformation is determined by an element of $V\left(\prod_{s \in S} D_s^{V(D_s)}\right)$ and the element that we want is the image of that element under the function induced by the projection:

$V\left(\prod_{s \in S} D_s^{V(D_s)}\right) \overset{V(p_{s_0,v})}{\to} V(D_{s_0})$

As $V$ is product-preserving, we can rearrange this to:

$\prod_{s \in S} V(D_s)^{V(D_s)} \overset{p_{s_0,v}}{\to} V(D_{s_0}).$

The projection selects the $(s_0,v)$th term of the element in question, and this element is, by definition, $v$. Hence the induced function on $V(D_{s_0})$ is the identity and the adjunction is shown.

Revised on January 1, 2012 05:21:07
by Mike Shulman
(173.8.161.189)