nLab
infinitary Lawvere theory

Idea

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. 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 $\kappa$-ary Lawvere theories for any regular cardinal $\kappa$.

Definitions

A many-sorted infinitary Lawvere theory is a locally small category $𝒟$ with all small products and with a small collection $ℛ$ 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 $𝒟$ with all small products and equipped with an $S$-indexed family $ℛ$ 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 $𝒟$ from the discrete category on $S$.)

If $(𝒟,ℛ)$ is an $S$-sorted infinitary Lawvere theory, then $𝒟$ 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 $ℛ$ and the $S$-indexed family is given by the identity indexing.

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

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 $(𝒟,R)$ 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.

Size issues

Freyd and Street (1995) have shown that a category $𝒟$ is small if and only if both $𝒟$ and the functor category $[𝒟,\mathrm{Set}]$ are locally small. Analogously, it seems that a category $𝒟$ with products may be given the structure of a many-sorted infinitary Lawvere theory if and only if both $𝒟$ and $\mathrm{Prod}[𝒟,\mathrm{Set}]$ (the category of product-preserving functors from $𝒟$ to $\mathrm{Set}$, a full subcategory of $[𝒟,\mathrm{Set}]$) 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.

Algebraic Categories

Conjecture

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

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

Lemma

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

Proof

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

As $S$ is a set, the product

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

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

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

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

$$d:D\cong \prod _{s\in S}{D}_s^{X_s}$$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 $p_{s,x}:{\prod }_{s\in S}{D}_s^{X_s}\to D_s$. As $V_1$ and $V_2$ are product-preserving, we have isomorphisms

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

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

Lemma

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

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

Proof

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

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

$$\eta :I\to UF,ϵ:FU\to I$$\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 $\mathrm{Set}$)

$${\eta }_{X,s_0}:X_{s_0}\to 𝒟(\prod _{s\in S}{D}_s^{X_s},D_{s_0}).$$\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:

$$\prod _{s\in S}{D}_s^{X_s}\to D_{s_0}^{X_{s_0}}\stackrel{p_x}{\to }{D}_{s_0}.$$\prod_{s \in S} D_s^{X_s} \to D_{s_0}^{X_{s_0}} \overset{p_x}{\to} D_{s_0}.

The counit, $ϵ$, is a little more complicated to describe. Let $V:𝒟\to \mathrm{Set}$ be a covariant product-preserving functor. Evaluated at $V$, ${ϵ}_V$ is a natural transformation

$$𝒟(\prod _{s\in S}{D}_s^{V(D_s)},-)\to V$$\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:

$$V\left(\prod _{s\in S}{D}_s^{V(D_s)}\right)\cong \prod _{s\in S}V(D_s{)}^{V(D_s)}$$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 ${ϵ}_V$.

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

$$F\stackrel{F\eta }{\to }FUF\stackrel{ϵF}{\to }F$$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

$$𝒟(\prod _{s\in S}{D}_s^{X_s},-)$$\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

$$𝒟(\prod _{s\in S}{D}_s^{X_s},\prod _{s\in S}{D}_s^{X_s})$$\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 ${\eta }_{X,s_0}:X_{s_0}\to 𝒟({\prod }_{s\in S}{D}_s^{X_s},D_{s_0})$ and thus a morphism

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

$$𝒟(\prod _{s\in S}{D}_s^{X_s},\prod _{s\in S}{D}_s^{X_s})\to 𝒟(\prod _{s\in S}{D}_s^{𝒟(\prod _{s\prime \in S}{D}_{s\prime }^{X_{s\prime }},D_s)},\prod _{s\in S}{D}_s^{X_s})$$\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 $ϵF$ to get a morphism:

$$𝒟(\prod _{s\in S}{D}_s^{𝒟(\prod _{s\prime \in S}{D}_{s\prime }^{X_{s\prime }},D_s)},\prod _{s\in S}{D}_s^{X_s})\to 𝒟(\prod _{s\in S}{D}_s^{X_s},\prod _{s\in S}{D}_s^{X_s})$$\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:

$$\begin{array}{ccccc}𝒟(\prod _{s\in S}{D}_s^{𝒟(\prod _{s\prime \in S}{D}_{s\prime }^{X_{s\prime }},D_s)},\prod _{s\in S}{D}_s^{𝒟(\prod _{s\prime \in S}{D}_{s\prime }^{X_{s\prime }},D_s)})& \stackrel{ϵ}{\to }& 𝒟(\prod _{s\in S}{D}_s^{X_s},\prod _{s\in S}{D}_s^{𝒟(\prod _{s\prime \in S}{D}_{s\prime }^{X_{s\prime }},D_s)})& \cong & \prod _{s\in S}𝒟{(\prod _{s″\in S}{D}_{s″}^{X_{s″}},D_s)}^{𝒟(\prod _{s\prime \in S}{D}_{s\prime }^{X_{s\prime }},D_s)}\\ \downarrow & & \downarrow & & \downarrow \\ 𝒟(\prod _{s\in S}{D}_s^{𝒟(\prod _{s\prime \in S}{D}_{s\prime }^{X_{s\prime }},D_s)},\prod _{s\in S}{D}_s^{X_s})& \stackrel{ϵ}{\to }& 𝒟(\prod _{s\in S}{D}_s^{X_s},\prod _{s\in S}{D}_s^{X_s})& \cong & \prod _{s\in S}𝒟{(\prod _{s″\in S}{D}_{s″}^{X_{s″}},D_s)}^{X_s}\end{array}$$\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 $ϵ$ 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 $ϵ$ 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:

$$U\stackrel{\eta U}{\to }UFU\stackrel{Uϵ}{\to }U$$U \overset{\eta U}{\to} U F U \overset{U \epsilon}{\to} U

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

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

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

$$V(D_{s_0})\to 𝒟(\prod _{s\in S}{D}_s^{V(D_s)},D_{s_0})\to V(D_{s_0}).$$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 $\eta$:

$$V(D_{s_0})\to 𝒟(\prod _{s\in S}{D}_s^{V(D_s)},D_{s_0})$$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 $ϵ$. This gives us a natural transformation

$$𝒟(\prod _{s\in S}{D}_s^{V(D_s)},-)\to V(-)$$\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:

$$V\left(\prod _{s\in S}{D}_s^{V(D_s)}\right)\stackrel{V(p_{s_0,v})}{\to }V(D_{s_0})$$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:

$$\prod _{s\in S}V(D_s{)}^{V(D_s)}\stackrel{p_{s_0,v}}{\to }V(D_{s_0}).$$\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.