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May 1, 2017

A Discussion on Notions of Lawvere Theories

Posted by Emily Riehl

Guest post by Daniel Cicala

The Kan Extension Seminar II continues with a discussion of the paper Notions of Lawvere Theory by Stephen Lack and Jirí Rosický.

In his landmark thesis, William Lawvere introduced a method to the study of universal algebra that was vastly more abstract than those previously used. This method actually turns certain mathematical stuff, structure, and properties into a mathematical object! This is achieved with a Lawvere theory: a bijective-on-objects product preserving functor T: 0 opLT \colon \aleph^{\text{op}}_0 \to \mathbf{L} where 0\aleph_0 is a skeleton of the category FinSet\mathbf{FinSet} and L\mathbf{L} is a category with finite products. The analogy between algebraic gadgets and Lawvere theories reads as: stuff, structure, and properties correspond respectively to 1, morphisms, and commuting diagrams.

To get an actual instance, or a model, of an algebraic gadget from a Lawvere theory, we take a product preserving functor m:TSetm \colon \mathbf{T} \to \mathbf{Set}. A model picks out a set m(1)m(1) and nn-ary operations m(f):m(1) nm(1)m(f) \colon m(1)^n \to m(1) for every TT-morphism f:n1f \colon n \to 1.

To read more about classical Lawvere theories, you can read Evangelia Aleiferi’s discussion of Hyland and Power’s paper on the topic.

With this elegant perspective on universal algebra, we do what mathematicians are wont to do: generalize it. However, there is much to consider undertaking such a project. Firstly, what elements of the theory ought to be generalized? Lack and Rosický provide a clear answer to this question. They generalize along the following three tracks:

  • consider a class of limits besides finite products,

  • replace the base category Set\mathbf{Set} with some other suitable category, and

  • enrich everything.

Another important consideration is to determine exactly how far to generalize. Why not just go as far as possible? Here are two reasons. First, there are a number of results in this paper that stand up to further generalization if one doesn’t care about constructibility. A second limiting factor of generalization is that one should ensure that central properties still hold. In Notions of Lawvere Theory, the properties lifted from classical Lawvere theories are

  • the correspondence between Lawvere theories and monads,

  • that algebraic functors have left adjoints, and

  • models form reflective subcategories of certain functor categories.

Before starting the discussion of the paper, I would like to take a moment to thank Alexander, Brendan and Emily for running this seminar. I have truly learned a lot and have enjoyed wonderful conversations with everyone involved.

Replacing finite limits

To find a suitable class of limits to replace finite products, we require the concept of presentability. The best entry point is to learn about local finite presentability, which David Myers has discussed here. With a little modification to the ideas there, we define a notion of local strong finite presentability and local Φ\Phi-presentability for a class of limits Φ\Phi.

We begin with sifted colimits, which are those Set\mathbf{Set}-valued colimits that commute with finite products. Note the familiarity of this definition with the commutativity property of filtered colimits. Of course, filtered colimits are also sifted. Another example is a reflexive pair, that is a category with shape


Anyway, we now look at the strongly finitely presentable objects in a category C\mathbf{C}. These are those objects xx whose representable C(x,):CSet\mathbf{C}(x,-) \colon \mathbf{C} \to \mathbf{Set} preserves sifted colimits. Denote the full subcategory of these by C sfp\mathbf{C}_{\text{sfp}}. Some simple examples include Set sfp\mathbf{Set}_{\text{sfp}}, which consists of the finite sets, and Ab sfp\mathbf{Ab}_{\text{sfp}}, which has the free and finitely generated Abelian groups. Also, given a category C\mathbf{C} of models for a Lawvere theory, C sfp\mathbf{C}_{\text{sfp}} is exactly those finitely presentable objects that are regular projective. A category C\mathbf{C} is locally strongly finitely presentable if it is cocomplete, C sfp\mathbf{C}_{\text{sfp}} is small, and any C\mathbf{C}-object is a sifted colimit of a diagram in C sfp\mathbf{C}_{\text{sfp}}. There is also a nice characterization (Theorem 3.1 in the paper) that states C\mathbf{C} is locally strongly finitely presentable if and only if C sfp\mathbf{C}_{\text{sfp}} has finite coproducts and we can identity C\mathbf{C} with the category of finite product-preserving functors C sfp opSet\mathbf{C}^{\text{op}}_{\text{sfp}} \to \mathbf{Set}. One of the most important results of Notions of Lawvere Theory, was in expanding the theory to encompass sifted (weighted) colimits. More on this later.

We can play this game with any class of limits Φ\Phi. Before defining PhiPhi-presentability, here is a bit of jargon.

Definition. A functor is Φ\Phi-flat if its colimit commutes with Φ\Phi-limits.

We call an object xx of a category C\mathbf{C} Φ\Phi-presentable if C(x,):CSet\mathbf{C}(x,-) \colon \mathbf{C} \to \mathbf{Set} preserves Φ\Phi-flat colimits. Given the full subcategory C Φ\mathbf{C}_\Phi of Φ\Phi-presentable objects, we call C\mathbf{C} locally Φ\Phi-presentable if it is cocomplete, C Φ\mathbf{C}_{\Phi} is small, and any C\mathbf{C}-object is a Φ\Phi-flat colimit of a diagram in C Φ\mathbf{C}_{\Phi}. Fortunately, we retain the characterization of C\mathbf{C} being locally Φ\Phi-presentable if and only if C Φ\mathbf{C}_{\Phi} has Φ\Phi-colimits and C\mathbf{C} is equivalent to the category Φ\Phi-Cts(C Φ op,Set)\mathbf{Cts}(\mathbf{C}^{\text{op}}_\Phi, \mathbf{Set}) of Φ\Phi-continuous functors C Φ opSet\mathbf{C}^{\text{op}}_\Phi \to \mathbf{Set}. Important results in Notions of Lawvere Theory use the assumption of Φ\Phi-presentability.

Let’s come back to Lawvere theories. From this point on, we fix a symmetric monoidal closed category 𝒱\mathcal{V} that is both complete and cocomplete. Also, Φ\Phi will refer to a class of weights over 𝒱\mathcal{V}. Our first task will be to determine what class of limits can replace finite products in the classical case. To this end, we take the following assumption.

Axiom A. Φ\Phi-continuous weights are Φ\Phi-flat.

This axiom is an analogy with how filtered colimits commute with finite limits in Set\mathbf{Set}. But for what classes of limits Φ\Phi does this hold?

To answer this question, we fix a sound doctrine 𝔻\mathbb{D}. Very roughly, a sound doctrine is a collection of small categories whose limits behave nicely with respect to certain colimits. After putting some small assumptions on the underlying category 𝒱 0\mathcal{V}_0 which we’ll sweep under the rug, define 𝒱 𝔻\mathcal{V}_\mathbb{D} to be the full sub 𝒱\mathcal{V}-category consisting of those objects xx such that [x,]:𝒱𝒱[x,-] \colon \mathcal{V} \to \mathcal{V} preserves 𝔻\mathbb{D}-flat colimits. Let Φ\Phi be the class of limits ‘built from’ conical 𝔻\mathbb{D}-limits and 𝒱 𝔻\mathcal{V}_\mathbb{D}-powers in the sense that we take ϕΦ\phi \in \Phi if

  • any 𝒱\mathcal{V}-category with conical 𝔻\mathbb{D} limits and 𝒱 𝔻\mathcal{V}_\mathbb{D}-powers also admits ϕ\phi-weighted limits, and

  • any 𝒱\mathcal{V}-functor conical 𝔻\mathbb{D} limits and 𝒱 𝔻\mathcal{V}_\mathbb{D}-powers also preserves ϕ\phi-weighted limits.

The fancy way of saying this is that Φ\Phi is the saturation class of conical 𝔻\mathbb{D}-limits and 𝒱 𝔻\mathcal{V}_\mathbb{D}-powers. It’s easy enough to see that Φ\Phi contains the conical 𝔻\mathbb{D}-limits and 𝒱\mathcal{V}-powers.

Having constructed a class of limits Φ\Phi from a sound doctrine 𝔻\mathbb{D}, we use the following theorem to imply that Φ\Phi satisfies the axiom above.

Theorem. Let 𝒦\mathcal{K} be a small V\mathbf{V}-category with Φ\Phi-weighted limits and F:𝒦𝒱F \colon \mathcal{K} \to \mathcal{V} be a V\mathbf{V}-functor. The following are equivalent:

  • FF is 𝔻\mathbb{D}-continuous;

  • FF is Φ\Phi-flat

  • FF is Φ\Phi-continuous.

In particular, the first item allows us to construct Φ\Phi using sound limits and the equivalence between the second and third item is precisely the axiom of interest. Here are some examples.

Example. Let 𝔻\mathbb{D} be the collection of all finite categories. We will also take 𝒱 0\mathcal{V}_0 be locally finitely presentable with the additional requirement that the monoidal unit II is finitely presentable as is the tensor product of two finitely presentable objects. Examples of such a 𝒱\mathcal{V} are categories of sets, abelian groups, modules over a commutative ring, chain complexes, categories, groupoids, and simplicial sets. Then Φ\Phi, as constructed from 𝔻\mathbb{D} above gives a good notion of 𝒱\mathcal{V}-enriched finite limits.

Example. A second example, and one of the main contributions of Notions of Lawvere Theory is when 𝔻\mathbb{D} is the class of all finite, discrete categories. Here, we take our 𝒱\mathcal{V} as in the first example, though we do not require the monoidal unit to be strongly finitely presentable. We do this because, by requiring the monoidal unit to be strongly finitely presentable, we lose the example where 𝒱\mathcal{V} is the category of directed graphs, which happens to be a key example, particularly to realizing categories as an model of a Lawvere theory. In this case, the induced class Φ\Phi gives an enriched version of strongly finite limits as discussed above. This Φ\Phi generalizes finite products in the sense that they coincide when 𝒱\mathcal{V} is Set\mathbf{Set}.

Correspondence between Lawvere theories and monads

Now that we’ve gotten our hands on some suitable limits, let’s see how we can obtain the classical correspondence between Lawvere theories and monads. Naturally, we’ll be assuming axiom A. In addition, we fix a 𝒱\mathcal{V}-category 𝒦\mathcal{K} that satisfies the following.

Axiom B1. 𝒦\mathcal{K} is locally Φ\Phi-presentable.

This axiom implies, as in our discussions above, that 𝒦Φ\mathcal{K} \cong \Phi-Cts(𝒦 Φ op,𝒱)\mathbf{Cts}(\mathcal{K}^{\text{op}}_\Phi,\mathcal{V}). This is not particularly restrictive, as presheaf 𝒱\mathcal{V}-categories are locally Φ\Phi-presentable.

Now, define a Lawvere Φ\Phi-theory on 𝒦\mathcal{K} to be a bijective-on-objects V\mathbf{V}-functor g:𝒦 Φ opg\colon \mathcal{K}^{\text{op}}_{\Phi} \to \mathcal{L} that preserves Φ\Phi-limits. A striking difference between a Lawvere Φ\Phi-theory and the classical notion is that the former does not require \mathcal{L} to have the limits under consideration. This makes defining the models of a Lawvere Φ\Phi-theory a subtler issue than in the classical case. Instead of defining a model to be a Φ\Phi-continuous functor as one might expect, we instead use the pullback square


To understand what a model looks like, use the intuition for a pullback in the category Set\mathbf{Set} and the fact that 𝒦\mathcal{K} is equivalent to Φ\Phi-Cts(𝒦 Φ op,𝒱)\mathbf{Cts}(\mathcal{K}^{\text{op}}_\Phi,\mathcal{V}). So a model will be a 𝒱\mathcal{V}-functor 𝒱\mathcal{L} \to \mathcal{V} whose restriction along gg is Φ\Phi-continuous.

The other major player in this section is the category of Φ\Phi-flat monads Mnd Φ(𝒦)\mathbf{Mnd}_{\Phi}(\mathcal{K}). We claim that there is an equivalence between Law Φ(𝒦)\mathbf{Law}_{\Phi}(\mathcal{K}) and Mnd Φ(𝒦)\mathbf{Mnd}_{\Phi}(\mathcal{K}). To verify this, we construct a pair of functors between Law Φ(𝒦)\mathbf{Law}_{\Phi}(\mathcal{K}) and Mnd Φ(𝒦)\mathbf{Mnd}_{\Phi}(\mathcal{K}). The first under consideration: mnd:Law Φ(𝒦)Mnd Φ(𝒦){mnd} \colon \mathbf{Law}_{\Phi}(\mathcal{K}) \to \mathbf{Mnd}_{\Phi}(\mathcal{K}). We define this with the help of the following proposition.

Proposition. The functor uu from the above pullback diagram is monadic via a Φ\Phi-flat monad tt.

Hence, a Φ\Phi-theory \mathcal{L} gives a monadic functor u:Mod()𝒦u \colon \mathbf{Mod}(\mathcal{L}) \to \mathcal{K} that yields a monad tt on 𝒦\mathcal{K}. Moreover, this monad preserves all the limits required to be an object in Mnd Φ(𝒦)\mathbf{Mnd}_\Phi (\mathcal{K}). So, define mnd()=t{mnd} (\mathcal{L}) = t.

Next, we define a functor th:Mnd Φ(𝒦)Law Φ(𝒦){th} \colon \mathbf{Mnd}_{\Phi}(\mathcal{K}) \to \mathbf{Law}_{\Phi}(\mathcal{K}). Consider a monad tt in Mnd Φ(𝒦)\mathbf{Mnd}_{\Phi}(\mathcal{K}). As per usual, tt factors through the Eilenberg-Moore category 𝒦𝒦 t\mathcal{K} \to \mathcal{K}^t which we precompose with the inclusion 𝒦 Φ𝒦\mathcal{K}_\Phi \to \mathcal{K} giving f:𝒦 Φ𝒦 mf \colon \mathcal{K}_\Phi \to \mathcal{K}^m. Now defining a 𝒱\mathcal{V}-category 𝒢\mathcal{G} that has objects from 𝒦 Φ\mathcal{K}_\Phi and 𝒢(x,y)=𝒦 m(fx,fy)\mathcal{G}(x,y) = \mathcal{K}^m(fx,fy), we factor ff


where \ell is bijective-on-objects and rr is full and faithful. This factorization is unique up to unique isomorphism. Define th(t)=𝒢 op\text{th} (t) = \mathcal{G}^{\text{op}}.

At this point, we have functors mnd:Law Φ(𝒦)Mnd Φ(𝒦) and th:Mnd Φ(𝒦)Law Φ(𝒦) \text{mnd} \colon \mathbf{Law}_{\Phi}(\mathcal{K}) \to \mathbf{Mnd}_{\Phi}(\mathcal{K}) \quad \text{ and } \quad \text{th} \colon \mathbf{Mnd}_{\Phi}(\mathcal{K}) \to \mathbf{Law}_{\Phi}(\mathcal{K}) so let’s turn our attention to showing that these are mutual weak inverses. The first step is to show that the category of algebras 𝒦 t\mathcal{K}^t for a given monad tt is the category of models Mod(th(t))\mathbf{Mod}(\text{th} (t)).

Theorem 6.6. The 𝒱\mathcal{V}-functor 𝒦 t(r,):𝒦 t[th(t) op,𝒱],x𝒦 m(r,x) \mathcal{K}^t(r-,-) \colon \mathcal{K}^t \to [\text{th} (t)^{\text{op}},\mathcal{V}], \quad x \mapsto \mathcal{K}^m(r-,x) restricts to an isomorphism of 𝒱\mathcal{V}-categories 𝒦 tMod(th(t))\mathcal{K}^t \cong \mathbf{Mod}(\text{th} (t)).

This theorem gives us that mndthid\text{mnd} \circ \text{th} \cong \text{id}. The next theorem gives us the other direction.

Theorem 6.7. There is an isomorphism thmndid\text{th} \circ \text{mnd} \cong \text{id}.

Let’s sketch the proof. Let g:𝒦 Φ opg \colon \mathcal{K}^{\text{op}}_{\Phi} \to \mathcal{L} be a Lawvere Φ\Phi-theory. If we denote mnd(𝒯)\text{mnd} (\mathcal{T}) by tt, we get thmnd(𝒯)=th(t)=𝒢 op\text{th} \circ \text{mnd} (\mathcal{T}) = \text{th} (t) = \mathcal{G}^{\text{op}} via the factorization


where \ell is bijective-on-objects and rr is fully faithful. It remains to show that 𝒯=𝒢 op\mathcal{T} = \mathcal{G}^{\text{op}}.
Let’s compute the image of an 𝒦 Φ\mathcal{K}_\Phi-object xx in Mod Φ(th(t))\mathbf{Mod}_{\Phi} (\text{th} (t)). For this, recall that we have 𝒦Φ\mathcal{K} \simeq \Phi-Cts(𝒦 Φ op,𝒱)\mathbf{Cts}(\mathcal{K}^{\text{op}}_{\Phi},\mathcal{V}) by assumption. Embedding xx into Φ\Phi-Cats(𝒦 Φ,𝒱)\mathbf{Cats}(\mathcal{K}_{\Phi},\mathcal{V}) gives us 𝒦 Φ(,x):𝒦 Φ op𝒱. \mathcal{K}_\Phi(-,x) \colon \mathcal{K}^{\text{op}}_{\Phi} \to \mathcal{V}. This, in turn, is mapped to the left Kan extension Lan g(𝒦 Φ(,x)):𝒯𝒱 \text{Lan}_g (\mathcal{K}_\Phi(-,x)) \colon \mathcal{T} \to \mathcal{V} along g:𝒦 Φ op𝒯g \colon \mathcal{K}^{\text{op}}_{\Phi} \to \mathcal{T} (the Lawvere Φ\Phi theory we began with). Here, we can compute that Lan g(𝒦 Φ(,x))\text{Lan}_g (\mathcal{K}_\Phi(-,x)) is 𝒯(,gx)\mathcal{T}(-,gx) meaning the factorization above is


Therefore, 𝒯=𝒢 op\mathcal{T} = \mathcal{G}^{\text{op}} as desired.

Many-sorted theories

Moving from single-sorted to many-sorted theories, we will take a different assumption on our 𝒱\mathcal{V}-category 𝒦\mathcal{K}.

Axiom B2. 𝒦\mathcal{K} is a 𝒱\mathcal{V}-category with Φ\Phi-limits and such that the Yoneda inclusion 𝒦[𝒦 op,𝒱]\mathcal{K} \to [\mathcal{K}^{\text{op}} , \mathcal{V}] has a Φ\Phi-continuous left adjoint.

This requirement on 𝒦\mathcal{K} is not overly restrictive as it holds for all presheaf 𝒱\mathcal{V}-categories and all Grothendieck topoi when 𝒱\mathcal{V} is Set\mathbf{Set}. The nice thing about this assumption is that we can compute all colimits and Φ\Phi-limits in 𝒦\mathcal{K} by passing to [𝒦 op,𝒱][\mathcal{K}^{\text{op}} , \mathcal{V}], where they commute, then reflecting back.

Generalizing Lawvere theories here is a bit simpler than in the previous section. Indeed, call any small 𝒱\mathcal{V}-category \mathcal{L} with Φ\Phi-limits a Φ\Phi-theory. Notice that we no longer have a bijective-on-objects functor involved in the definition. That functor forced the single-sortedness. With the functor no longer constraining the structure, we have the possibility for many sorts. Also, a Φ\Phi-theory does have all Φ\Phi-limits here, unlike in the single-sorted case. This allows for a much simpler definition of a model. Indeed, the category of models for a Φ\Phi-theory LL is the full subcategory Φ\Phi-Cts(,𝒦)\mathbf{Cts}(\mathcal{L}, \mathcal{K}) of [,𝒦][\mathcal{L}, \mathcal{K}].

Presently, we are interested in generalizing two important properties of Lawvere theory to Φ\Phi-theories. The first is that algebraic functors have left adjoints. The second is the reflectiveness of models.

Algebraic functors have left adjoints. A morphism of Φ\Phi-theories is a Φ\Phi-continuous 𝒱\mathcal{V}-functor g:g \colon \mathcal{L} \to \mathcal{L}\prime. Any such morphism induces a pullback 𝒱\mathcal{V}-functor g *:Φg^{\ast} \colon \Phi-Cts(,𝒦)Φ\mathbf{Cts}(\mathcal{L}\prime, \mathcal{K}) \to \Phi-Cts(,𝒦)\mathbf{Cts}(\mathcal{L}, \mathcal{K}) between model 𝒱\mathcal{V}-categories. We call such functors Φ\Phi-algebraic. And yes, these do have left adjoints just as in the context of classical Lawvere theories.

Theorem. Let \mathcal{L} and \mathcal{L}\prime be Φ\Phi-theories and g:g \colon \mathcal{L} \to \mathcal{L}\prime a 𝒱\mathcal{V}-functor between them. Given a model m:𝒦m \colon \mathcal{L} \to \mathcal{K}, then Lan gm:𝒦\text{Lan}_g m \colon \mathcal{L}\prime \to \mathcal{K} is a model.

What is happening here? Of course, pulling back by gg gives a way to turn models of \mathcal{L}\prime into models of \mathcal{L} – this is the algebraic functor g *g^{\ast}. But the left Kan extension along gg gives a way to turn a model mm of \mathcal{L} into a model of \mathcal{L}\prime as depicted in the diagram


This theorem says that process gives a functor g *:Φg_{\ast} \colon \Phi-Cts(,𝒦)Φ\mathbf{Cts}(\mathcal{L}, \mathcal{K}) \to \Phi-Cts(,𝒦)\mathbf{Cts}(\mathcal{L}\prime, \mathcal{K}) given by mLan gmm \mapsto \text{Lan}_g m.

We can prove this theorem for 𝒦=𝒱\mathcal{K} = \mathcal{V} without requiring axiom B2. This axiom is used to extend this result to a 𝒱\mathcal{V}-category 𝒦\mathcal{K}. The existence of the left adjoint \ell to the Yoneda embedding yy of 𝒦\mathcal{K} gives a factorization Lan gm=Lan gym\text{Lan}_g m = \ell \text{Lan}_g y m. The proof then reduces to showing that Lan gym\text{Lan}_g y m is Φ\Phi-continuous, since we are already assuming that \ell is. But because the codomain of Lan gym\text{Lan}_g y m is [𝒦 op,𝒱][\mathcal{K}^{\text{op}},\mathcal{V}], we can rest on the fact that we have proven the result for 𝒦=𝒱\mathcal{K} = \mathcal{V}. Limits are taken pointwise, after all. Actually, the left adjoint to g *g^{\ast} holds more generally, but our assumptions on 𝒦\mathcal{K} allow us to explicitly compute the left adjoint with left Kan extensions.

Reflexiveness of models. Having discussed left adjoints of algebraic functors, we now move on to show that categories of models Φ\Phi-Cts(𝒯,𝒱)\mathbf{Cts}(\mathcal{T},\mathcal{V}) are reflexive in [𝒯,𝒱][\mathcal{T}, \mathcal{V}]. Consider the free-forgetful (ordinary) adjunction


between 𝒱\mathcal{V}-categories and those 𝒱\mathcal{V}-categories with Φ\Phi-limits and functors preserving them. Given a 𝒱\mathcal{V}-category \mathcal{L} in the image of UU. Note that \mathcal{L} is a Φ\Phi-theory. It follows from this adjunction that Φ\Phi-Cts(ℱℒ,𝒱)\mathbf{Cts}(\mathcal{FL},\mathcal{V}) is equivalent to the category [,𝒱][\mathcal{L},\mathcal{V}]. Moreover, since \mathcal{L} has Φ\Phi-limits, the inclusion ℱℒ\mathcal{L} \hookrightarrow \mathcal{FL} has a right adjoint RR inducing an algebraic functor R *:ΦR^{\ast} \colon \Phi-Cts(,𝒱)Φ\mathbf{Cts}(\mathcal{L},\mathcal{V}) \to \Phi-Cts(ℱℒ,𝒱)[,𝒱]\mathbf{Cts}(\mathcal{FL},\mathcal{V}) \simeq [\mathcal{L},\mathcal{V}]. But we just showed that algebraic functors have left adjoints, giving us the following theorem.

Theorem. Φ\Phi-Cts(,𝒱)\mathbf{Cts}(\mathcal{L},\mathcal{V}) is reflective in [,𝒱][\mathcal{L},\mathcal{V}].

As promised, in the two general contexts corresponding to the axioms B1 and B2, we have the Lawvere theory-monad correspondence, that algebraic functors have left adjoints, and that categories of models are reflective.

An example

After all of that abstract nonsense, let’s get our feet back on the ground. Here is an example of manifesting a category with a chosen terminal object as a generalized Lawvere theory. This comes courtesy of Nishizawa and Power.

Let 0\mathbf{0} denote the empty category, 1\mathbf{1} the terminal category, and 2\mathbf{2} the category {ab}\{a \to b\} with two objects and a single arrow between them. We will also take 𝒦=𝒱=Cat\mathcal{K} = \mathcal{V} = \mathbf{Cat}. The class of limits Φ\Phi here is the finite Cat\mathbf{Cat}-powers. We define a Lawvere Φ\Phi-theory \mathcal{L} to be the Cat\mathbf{Cat}-category where we formally add to Cat fp op\mathbf{Cat}^{\text{op}}_{\text{fp}} (the opposite full subcategory on the finitely presentable objects) two arrows: τ:01\tau \colon \mathbf{0} \to \mathbf{1} and σ:12\sigma \colon \mathbf{1} \to \mathbf{2}. We then close up under finite Cat\mathbf{Cat}-powers and modulo the commutative diagrams

diagram8 \quad \quad diagram9 \quad \quad diagram10

Now \mathcal{L} is the Lawvere Φ\Phi-theory for a category with a chosen terminal object. A model of \mathcal{L} is a Φ\Phi-continuous Cat\mathbf{Cat}-functor Cat\mathcal{L} \to \mathbf{Cat}. This means that if MM is a model, it must preserve powers and so the following diagrams must commute:

diagram11 \quad \quad diagram12 \quad \quad diagram13

Here dom\text{dom} and cod\text{cod} choose the domain and codomain, and Δ\Delta the diagonal functor. Note that the commutativity of these diagrams witnesses the preservation of raising the terminal category to the first three diagrams given. Let’s parse these diagrams out.

The category we get from the model MM is M1M1 and the distinguished terminal object tt is chosen by MτM\tau. The first two diagrams provide a morphism xtx \to t for every object xx in M1M1. The third diagram gives the identity map on tt. The uniqueness of maps into tt follows from the functorality of MσM\sigma and cod\text{cod}.

Conversely, given a category C\mathbf{C} with a chosen terminal object tt, define a model M:CatM \colon \mathcal{L} \to \mathbf{Cat} by 1C\mathbf{1} \to \mathbf{C} and 1 xC x\mathbf{1}^x \to \mathbf{C}^x for \mathcal{L}-objects xx. Also, let MτM\tau choose tt and let MσM \sigma send each \mathcal{L}-object xx to the !:xt! \colon x \to t.

Posted at May 1, 2017 11:37 PM UTC

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Re: A discussion on notions of Lawvere theories

Thanks for a very nice refactoring of an already nice paper.

Back when we were discussing Lawvere theories vs. monads, I got the impression that ‘finiteness’ was the main difference between monads coming from theories and arbitrary monads. But after reading your post and David’s, I now see that ‘preservation of (co)limits’ would be a better description of the difference.

(And ultimately, we can’t talk about finiteness without talking about preservation of limits)

Thanks for the example of a Lawvere theory for categories with a terminal object. In the paper you linked, they also give theories for categories with binary products and cartesian closed categories.

I was wondering if we can talk about more complicated things like abelian categories in this framework. The part about being an Ab\mathbf{Ab}-enriched category with all finite limits and finite colimits is easy. But I can’t see how one would then encode the condition that all monos are kernels and all epis are cokernels.

Posted by: Ze on May 3, 2017 9:12 AM | Permalink | Reply to this

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