### Classifying by Generalizing: The Theory of Accessibility Relative to a Limit Doctrine

#### Posted by Emily Riehl

*Guest post by Tim Campion*

In the tenth installment of the Kan Extension Seminar, we take a break from all the 2-category theory we’ve been doing, in favor of some good old-fashioned 1-category theory (although if you keep an eye out, you may notice some higher dimensions lurking in the background!). This week’s paper is *A Classification of Accessible Categories* by Adámek, Borceux, Lack, and Rosický, which both generalizes and refines the classical theory of locally presentable and accessible categories by working relative to a new parameter, $\mathbb{D}$, called a *limit doctrine*. But don’t worry if you know nothing about locally presentable and accessible categories – I hope this post can serve as an introduction, from the $\mathbb{D}$-relative perspective.

The classical theory of locally presentable and accessible categories is notable partly because of its scope: many categories of intrinsic mathematical interest – from groups, rings, and fields, to categories themselves, to Banach spaces and contractive maps – are accessible. At the same time, the theory is notable for the richness of the categorical concepts that it brings together – from colimit completions, to sketches, to orthogonality classes. Both of these aspects will also be apparent in the generalized theory. We will recover, for example, not only the notion of a (multi sorted) Lawvere theory, but also much of the theory surrounding it.

I’d like to thank all my fellow seminar participants for thoughtful discussions so far, as well as everyone who’s joined the discussion at the Café. It’s been a fantastic experience , “popping open” the vacuum in which I’ve done category theory in the past! A special thanks to Emily for organizing all of this and providing feedback and support, and to Joe Hannon for what’s been a fruitful reading collaboration.

### $\mathbb{D}$-Filtered Colimits

The whole business rests on the choice of a small set $\mathbb{D}$ of small categories which we refer to as *the doctrine of $\mathbb{D}$-limits*, because we’re thinking of the categories $\mathcal{D} \in \mathbb{D}$ as indices for limit diagrams. For most of the results of the paper we require $\mathbb{D}$ to satisfy a technical condition called *soundness*; more on that later. Some examples to have in mind are

The doctrine $\mathbf{FIN}$ of finite limits, consisting of all categories with finitely many morphisms.

More generally, the doctrine $\lambda\mathbf{-LIM}$ of $\lambda$-small limits for some regular cardinal $\lambda$, consisting of all categories with fewer than $\lambda$ morphisms. So $\mathbf{FIN}$ is $\aleph_0\mathbf{-LIM}$.

The doctrine $\mathbf{FINPR}$ of finite products, consisting of all finite discrete categories.

Also worth considering are

The doctrine $\mathbf{FINCL}$ of finite connected limits, consisting of all finite connected categories.

The empty doctrine $\emptyset$ consisting of no categories. (This is 0$\mathbf{-LIM}$.)

The doctrine $\mathbf{TERM}$ of the terminal object, consisting of the empty category. (This is $1\mathbf{-LIM}$.)

The classical case of $\lambda$-accessible categories will be recovered if we choose $\mathbb{D}=\lambda\mathbf{-LIM}$.

Recall that for small categories $\mathcal{C}, \mathcal{D}$, we say that *$\mathcal{D}$-limits commute with $\mathcal{C}$-colimits in $\mathbf{Set}$* if, for every functor $F: \mathcal{C} \times \mathcal{D} \to \mathbf{Set}$, the canonical map

$\operatorname{colim}_{c \in \mathcal{C}} \operatorname{lim}_{d \in \mathcal{D}} F(c,d) \to \operatorname{lim}_{d \in \mathcal{D}} \operatorname{colim}_{c \in \mathcal{C}} F(c,d)$

is an isomorphism. We now say that a small category $\mathcal{C}$ is *$\mathbb{D}$-filtered* if $\mathcal{D}$-limits commute with $\mathcal{C}$-colimits for every $\mathcal{D} \in \mathbb{D}$. In our examples,

The $\mathbf{FIN}$-filtered categories are usually just called filtered. A category $\mathcal{C}$ is filtered iff there is a cone on every finite diagram in $\mathcal{C}$.

The $\lambda\mathbf{-LIM}$-filtered categories are usually just called

*$\lambda$-filtered*. A category $\mathcal{C}$ is $\lambda$-filtered iff there is a cone on every diagram in $\mathcal{C}$ with $\lt\lambda$ morphisms.The $\mathbf{FINPR}$-filtered categories are called sifted (en français:

*tamisante*). Filtered categories are sifted, and so are the index category for reflexive coequalizers (two arrows with a common section) and the co-simplex category $\Delta^\mathrm{op}$.The $\mathbf{FINCL}$-filtered categories are the coproducts of filtered categories.

Every category is $\emptyset$-filtered.

The $\mathbf{TERM}$-filtered categories are the connected categories.

Finally, an object $x$ of a category $\mathcal{K}$ is said to be *$\mathbb{D}$-presentable* if the covariant hom-functor $\mathcal{K}(x,-): \mathcal{K} \to \mathbf{Set}$ preserves $\mathbb{D}$-filtered colimits. We will shortly see some examples of what this means for various $\mathbb{D}$ and $\mathcal{K}$.

### Locally $\mathbb{D}$-Presentable Categories

We could dive straight into the theory of $\mathbb{D}$-accessible categories, but the most important aspects of the theory are already present when we look at the *locally $\mathbb{D}$-presentable categories*, which are just the $\mathbb{D}$-accessible categories which are also cocomplete. Some simplifications are also possible in the locally $\mathbb{D}$-presentable setting, and lots of important examples are already encompassed. So let’s start with this case, and later discuss how the theory is modified in the more general case.

**Definition 1:** A locally $\mathbb{D}$-presentable category is a cocomplete category $\mathcal{K}$ with a small full subcategory $\mathcal{A}$ of $\mathbb{D}$-presentable objects such that every object is a $\mathbb{D}$-filtered colimit of objects of $\mathcal{A}$.

A locally $\emptyset$-presentable category $\mathcal{K}$ is just a category $[\mathcal{A}^\op, \mathbb{Set}]$ of presheaves on a small category $\mathcal{A}$. A presheaf is $\emptyset$-presentable iff it is a retract of a representable.

A locally $\mathbf{FINPR}$-presentable category is just the category of models for a (possibly multi-sorted) Lawvere theory: these are sometimes called the algebraic categories, or the many-sorted varieties. This includes the locally $\emptyset$-presentable categories, as well as the categories of Groups, Rings, $R$-Modules, Lattices, etc. The $\mathbf{FINPR}$-presentable objects of any of these categories are the retracts of the free algebras.

A locally $\mathbf{FIN}$-presentable category is simply called locally finitely presentable. This includes all locally $\mathbf{FINPR}$-presentable categories, as well as the category of Small Categories, the category of Posets, and many others. In these examples (including e.g. Categories and $R$-Modules), a finitely presentable object is one which is finitely presentable in the usual sense: it is finitely generated, and subject to finitely many equations.

A locally $\lambda\mathbf{-LIM}$-presentable category is called locally $\lambda$-presentable. In addition to the locally finitely presentable categories, these include the category of $C^{\ast}$-Algebras, the category of Banach spaces and contractive maps (both locally $\aleph_1$-presentable) and other categories of algebras with infinitary operations, as well as any Grothendieck topos. And $\lambda$-presentability is analogous to finite presentability.

On the other hand, the category $\mathbf{Top}$ of topological spaces, for example, is not locally $\mathbb{D}$-presentable for any $\mathbb{D}$: we will see later that every locally $\mathbb{D}$-presentable category is locally $\lambda$-presentable for some $\lambda$, and the $\lambda$-presentable objects of $\mathbf{Top}$ are the discrete spaces with $\lt\lambda$ points (as discussed here and here); their colimits are again discrete. The category of Hilbert spaces is also not locally $\mathbb{D}$-presentable because it is self dual, and the opposite of a locally $\mathbb{D}$-presentable category is never $\mathbb{D}$-presentable.

Also, beware that the “hereditary” nature of the locally $\mathbb{D}$-presentable world, where if $\mathbb{D} \subseteq \mathbb{D}'$ then locally $\mathbb{D}$-presentable categories are locally $\mathbb{D}'$-presentable, does not carry over so nicely to the $\mathbb{D}$-accessible case.

### … as $\mathbb{D}$-Filtered Cocompletions

Part of the power of the theory of locally $\mathbb{D}$-presentable and $\mathbb{D}$-accessible categories is that they are robust concepts admitting many equivalent definitions. Let’s look at a few of them.

The first one is of a very formal flavor. Consider the 2-category $\mathbf{\mathbb{D}-Fil-Cocts}$ of categories with all $\mathbb{D}$-filtered colimits, functors preserving $\mathbb{D}$-filtered colimits, and natural transformations. There is an obvious forgetful 2-functor $U: \mathbf{\mathbb{D}-Fil-Cocts} \to \mathbf{Cat}$, which has a left 2-adjoint which we will call $\mathbb{D}\mathbf{-Ind}$, so that $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$ is the *free $\mathbb{D}$-filtered cocompletion* of the category $\mathcal{A}$.

**Definition 2:** A locally $\mathbb{D}$-presentable category $\mathcal{K}$ is a free $\mathbb{D}$-filtered cocompletion $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$ of a small $\mathbb{D}^{\mathrm{op}}$-cocomplete category $\mathcal{A}$.

It’s a little funny that $\mathbf{\mathbb{D}-Ind}$ only adjoins $\mathbb{D}$-filtered colimits, and yet we end up with a cocomplete category. This comes down to the proviso that $\mathcal{A}$ be $\mathbb{D}^{\mathrm{op}}$-cocomplete (meaning that every diagram in $\mathcal{A}$ with index $\mathcal{D}^{\mathrm{op}}$ for some $\mathcal{D} \in \mathbb{D}$ has a colimit). This proviso will be lifted when we generalize to $\mathbb{D}$-accessible categories.

In order to relate Definition 2 to Definition 1, we must describe $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$ explicitly. It’s well-known that the free cocompletion of a small category $\mathcal{A}$ is its category of presheaves $[\mathcal{A}^{\mathrm{op}}, \mathbf{Set}]$ – that is, $\mathbf{\emptyset-Ind}(\mathcal{A}) = [\mathcal{A}^{\mathrm{op}}, \mathbf{Set}]$. It turns out that restricted cocompletions like $\mathbf{\mathbb{D}-Ind}$ can be formed similarly: for any doctrine $\mathbb{D}$ and small category $\mathcal{A}$, $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$ is the closure of the representables in $[A^{\mathrm{op}}, \mathbf{Set}]$ under $\mathbb{D}$-filtered colimits. When the doctrine $\mathbb{D}$ is *sound* (more on this later!), this colimit completion can be taken in one step – that is, a $\mathbb{D}$-filtered colimit of $\mathbb{D}$-filtered colimits of representables is a $\mathbb{D}$-filtered colimit of representables. This allows us to conclude that Definition 2 is equivalent, for a sound doctrine, to

**Definition 3:** A locally $\mathbb{D}$-presentable category $\mathcal{K}$ is a full subcategory of $[\mathcal{A}^{\mathrm{op}}, \mathbf{Set}]$, for some small, $\mathbb{D}^{\mathrm{op}}$-cocomplete category $\mathcal{A}$, consisting of the $\mathbb{D}$-filtered colimits of representables.

Using this description, we can at least state the relationship between Definition 1 and Definitions 2 and 3. If $\mathcal{K}$ is locally $\mathbb{D}$-presentable in the sense of Definition 1, then the nerve of the inclusion $i: \mathcal{K}_{\mathbb{D}} \to \mathcal{K}$ of the full subcategory $\mathcal{K}_{\mathbb{D}}$ of $\mathbb{D}$-presentables (which is essentially small and $\mathbb{D}^{\mathrm{op}}$-cocomplete )

$\begin{matrix} \mathcal{K} &\to &\mathbf{\mathbb{D}-Ind}(\mathcal{K}_{\mathbb{D}}) \\ K &\mapsto &\mathcal{K}(i-, K) \end{matrix}$

is an equivalence. Conversely, in $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$ the $\mathbb{D}$-presentable objects are the representables, and everything is a $\mathbb{D}$-filtered colimit of them.

### … as Categories of Continuous Functors

Definition 3 gives us an explicit description of a locally $\mathbb{D}$-presentable category as a category of presheaves, but it would be better to have a more *intrinsic* characterization of exactly which presheaves on $\mathcal{A}$ lie in $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$. The answer is very nice: a presheaf $F: \mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$ is a $\mathbb{D}$-filtered colimit of representables if and only if $F$ preserves $\mathcal{D}$-indexed limits for all $\mathcal{D} \in \mathbb{D}$. This yields

**Definition 4:** A locally $\mathbb{D}$-presentable category is a category consisting of all $\mathbb{D}$-continuous presheaves $F: \mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$ for some small $\mathbb{D}^{\mathrm{op}}$-cocomplete category $\mathcal{A}$.

A particularly nice example of this comes with $\mathbb{D}=\mathbf{FINPR}$. In this case, $\mathcal{A}^{\mathrm{op}}$ is a multi sorted Lawvere theory, and the associated locally $\mathbf{FINPR}$-presentable category is exactly the category of models of the Lawvere theory.

More generally, this correspondence between $\mathbb{D}$-complete categories like $\mathcal{A}^{\mathrm{op}}$ and the associated locally $\mathbb{D}$-presentable categories extends to a duality of 2-categories with the proper definitions. In the case $\mathbb{D} = \mathbf{FIN}$, this is known as Gabriel-Ulmer duality. The $\mathbb{D}$-relative case is due to Centazzo and Vitale.

A variation on this theme is to represent a locally $\mathbb{D}$-presentable category as a category of $\mathbf{Set}$-valued functors preserving not *all* $\mathbb{D}$-limits, but just *certain* ones. This is the theory of $\mathbb{D}$-sketches, and it is in the spirit of the Freyd-Kelly paper we read a couple of months ago, Categories of Continuous Functors.

A *limit $\mathbb{D}$-sketch* consists of a small category $\mathcal{T}$ and a set of cones in $\mathcal{T}$ indexed by categories in $\mathbb{D}$. A *model* of the sketch consists of a functor $\mathcal{T} \to \mathbf{Set}$ sending the designated cones to limit cones. A morphism of models is a natural transformation between them. We have

**Definition 5:** A locally $\mathbb{D}$-presentable category is the category of models of a limit $\mathbb{D}$-sketch.

Of course, Definition 4 implies Definition 5 by taking $\mathcal{T} = \mathcal{A}^{\mathrm{op}}$ and designating every limit cone with index in $\mathbb{D}$. The converse follows by completing $\mathcal{T}$ under $\mathbb{D}$-limits in the opposite of its category of models.

The advantage of using a sketch is that the category $\mathcal{T}$ can be smaller and more manageable than it has to be otherwise.

For example, when $\mathbb{D} = \mathbf{FINPR}$, the category of groups can be presented as the category of all finite-product preserving functors $\mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$ where $\mathcal{A}^{\mathrm{op}}$, the Lawvere theory for groups, is the opposite of the category of all finitely-generated free groups. But if we use a sketch, we can take $\mathcal{T}$ to consist just of the free groups on 0, 1, 2, and 3 generators, with appropriate product diagrams indicated: thus we essentially define a group to be a group object in $\mathbf{Set}$.

This is even more apparent when $\mathbb{D} = \mathbf{FIN}$. For example, to represent $\mathbf{Cat}$ as a category of finitely-continuous functors $\mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$, we must take $\mathcal{A}^\mathrm{op}$ to be *all* finitely presentable categories, which is rather awkward. But if we use a sketch, then we can take the category of simplices $\mathcal{T} = \Delta^{\mathrm{op}}$, with pullback diagrams indicating Segal conditions. Thus categories are represented by their simplicial nerves. We could even get away with $\mathcal{T}$ consisting just of the 0,1,2, and 3-dimensional simplices – then the sketch corresponds pretty directly to the usual notion of a internal category.

### … as $\mathbb{D}$-Orthogonality Classes

From the Freyd-Kelly paper, we know that categories of continuous functors can be thought of as orthogonality classes. And it turns out that locally $\mathbb{D}$-presentable categories can be recognized by the orthogonality classes defining them. We say that a $\mathbb{D}$-orthogonality class in a category $\mathcal{L}$ is a full subcategory of $\mathcal{L}$ consisting of the objects orthogonal to the arrows in a set $\mathcal{M}$, where the domains and codomains of arrows in $\mathcal{M}$ are $\mathbb{D}$-presentable. Then we have

**Definition 6:** The locally $\mathbb{D}$-presentable categories $\mathcal{K}$ are precisely the $\mathbb{D}$-orthogonality classes in presheaf categories.

More generally, a $\mathbb{D}$-orthogonality class in a locally $\mathbb{D}$-presentable category is always locally $\mathbb{D}$-presentable. These orthogonality classes are substantially simpler than the ones considered in the Freyd-Kelly paper: they are determined by a small set of arrows, and they are in functor categories with values in $\mathbf{Set}$ rather than a more general category.

### $\mathbb{D}$-Accessible Categories

Now it’s time to relax the cocompleteness condition for locally $\mathbb{D}$-presentable categories and see where the chips fall. Rather than *all* small colimits, a $\mathbb{D}$-accessible category is required only to have $\mathbb{D}$-filtered colimits.

**Definition 1’:** A $\mathbb{D}$-accessible category $\mathcal{K}$ is a category with $\mathbb{D}$-filtered colimits and a small set $\mathcal{A}$ of $\mathbb{D}$-presentable objects of which every object is a $\mathbb{D}$-filtered colimit.

So the locally $\mathbb{D}$-presentable categories are just the cocomplete $\mathbb{D}$-accessible categories. What does this added generality buy us?

The $\emptyset$-accessible categories are still just the presheaf categories.

The $\mathbf{FINPR}$-accessible categories have been called generalized varieties. Besides the varieties, they include the algebraic categories, but also the category of fields and the category of linearly ordered sets.

The $\mathbf{FIN}$-accessible categories are just called finitely accessible. Beyond the locally finitely presentable case, finitely accessible posets are known as continuous posets, and they are important in Domain theory.

The $\mathbf{\lambda-\mathbf{LIM}}$-accessible categories are just called $\lambda$-accessible. They include the locally $\lambda$-presentable categories; a non-cocomplete example is the category of Hilbert Spaces and linear contractions, which is countably accessible.

The $\mathbf{FINCL}$-accessible categories are just the free coproduct completions $\mathbf{Fam}(\mathcal{K})$ of accessible categories $\mathcal{K}$.

And how do the other equivalent definitions change? Definitions 2 and 3 actually simplify in the general case:

**Definition 2’:** A $\mathbb{D}$-accessible category $\mathcal{K}$ is the free $\mathbb{D}$-filtered cocompletion $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$ of a small category $\mathcal{A}$.

**Definition 3’:** A $\mathbb{D}$-accessible category $\mathcal{K}$ is a full subcategory of $[\mathcal{A}^{\mathrm{op}}, \mathbf{Set}]$, for some small category $\mathcal{A}$, consisting of the $\mathbb{D}$-filtered colimits of representables.

The change is that $\mathcal{A}$ can be an arbitrary small category; it is not required to be $\mathbb{D}^\mathrm{op}$-cocomplete.

When we come to Definition 4, trying to provide an intrinsic characterization of the presheaves of $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$, we hit a snag. If $\mathcal{A}$ is not $\mathbb{D}^\mathrm{op}$-cocomplete, then it doesn’t make much sense to consider $\mathbb{D}$-continuous functors $\mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$, and in fact $\mathbb{D}$-continuity is not a strong enough condition to single out the presheaves in $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$. Instead, we call the presheaves of $\mathbf{\mathbb{D}-Ind}(\mathcal{A})$ the *$\mathbb{D}$-flat functors*, in analogy to the flat functors which are recovered as the $\mathbf{FIN}$-flat functors. The condition of soundness allows us to characterize these functors in a few different ways, in analogy to the classical $\mathbb{D}=\mathbf{FIN}$ case:

**Theorem:** (2.4 in the paper) If $\mathbb{D}$ is sound and $F: \mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$ is a presheaf on a small category, then the following are equivalent:

The left Kan extension $\operatorname{Lan}_Y F : [A, \mathbf{Set}] \to \mathbf{Set}$ of $F$ along the Yoneda embedding preserves $\mathbb{D}$-limits of representables.

$\operatorname{Lan}_Y F$ is $\mathbb{D}$-continuous.

$F$ is a $\mathbb{D}$-filtered colimit of representables (i.e. $F \in \mathbf{\mathbb{D}-Ind}(\mathcal{A})$).

The category of elements $\operatorname{Elts}(F)^\mathrm{op}$ is $\mathbb{D}$-filtered.

These conditions collectively are taken to provide an adequate characterization of the $\mathbb{D}$-flat functors. If $\mathcal{A}$ is $\mathbb{D}$-cocomplete, $\mathbb{D}$-flatness reduces to $\mathbb{D}$-continuity.

**Definition 4’:** A $\mathbb{D}$-accessible category $\mathcal{K}$ is a category of $\mathbb{D}$-flat functors on a small category $\mathcal{A}$.

I should emphasize that this is not the only place where soundness comes up in the theory – just the most obvious.

Moving on, the theory of $\mathbb{D}$-sketches generalizes to the $\mathbb{D}$-accessible case. In addition to indicating certain $\mathbb{D}$-indexed cones to be sent to limits, we must allow ourselves to also specify certain colimit cones, of arbitrary index. Sketches provide a nice way to present $\mathbb{D}$-accessible categories. For example, the theory of fields can be sketched by sketching a ring $R$ and then indicating a decomposition of $R$ as the coproduct of two objects, one of which is equipped with an inverse operation, and the other of which contains only the zero element of the ring; see, for example here.

Unfortunately, though, there are $\mathbb{D}$-sketchable categories which are *not* $\mathbb{D}$-accessible. Examples exist even for $\mathbb{D} = \mathbf{FIN}$ and $\mathbb{D} = \mathbf{FINPR}$. It is a fact, though, that every $\mathbb{D}$-sketchable category is $\lambda$-accessible for some $\lambda$ (this is how we know that $\mathbb{D}$-accessible categories are accessible, as I alluded to earlier), but it is not possible to make general conclusions within a given doctrine $\mathbb{D}$.

**Non-Definition 5’:** All $\mathbb{D}$-accessible categories are $\mathbb{D}$-sketchable, but not conversely.

Finally, $\mathbb{D}$-accessible categories are not cocomplete outside the locally $\mathbb{D}$-presentable case, so they are clearly not reflective in presheaf categories. So Definition 6 doesn’t really have an analogue for $\mathbb{D}$-accessible categories.

### A Word About Soundness

The technical notion of *soundness* permeates this paper, but I’ve been trying to suppress it for the most part up to now. To define what it means for a doctrine $\mathbb{D}$ to be sound, let me first say what it means for a category to be *representably $\mathbb{D}$-filtered*:

Recall that a category $\mathcal{C}$ is $\mathbb{D}$-filtered if for every $\mathcal{D} \in \mathbb{D}$, and every $F: \mathcal{D} \times \mathcal{C} \to \mathbf{Set}$,
$\mathrm{colim}\,\mathrm{lim}\, F \cong \mathrm{lim}\,\mathrm{colim}\, F$ canonically. Similarly, $\mathcal{C}$ is *representably $\mathbb{D}$-filtered* if this condition holds for $F$ of the form $C(S,1)$ for some functor $S: \mathcal{D}^\mathrm{op} \to \mathcal{C}$. (If we think of $F$ as a profunctor $\mathcal{D}^\mathrm{op} \to \mathcal{C}$, these are the representable profunctors).

Certainly $\mathbb{D}$-filtered categories are representably $\mathbb{D}$-filtered. The doctrine $\mathbb{D}$ is called *sound* if the converse holds, so that $\mathbb{D}$-filteredness coincides with representable $\mathbb{D}$-filteredness.

One nice thing about soundness is that it makes $\mathbb{D}$-filteredness easier to check, because representable $\mathbb{D}$-filteredness admits a nice combinatorial description. In fact, for an arbitrary doctrine $\mathbb{D}$ and small category $\mathcal{C}$, the following are equivalent:

For each $\mathcal{D} \in \mathbb{D}$, every functor $S: \mathcal{D}^\mathrm{op} \to \mathcal{C}$ has a connected category of cocones.

For each $\mathcal{D} \in \mathbb{D}$, the diagonal functor $\Delta: \mathcal{C} \to [\mathcal{D}^\mathrm{op}, \mathcal{C}]$ is a final functor.

$\mathcal{C}$ is representably $\mathbb{D}$-filtered.

The functor $\mathrm{colim}: [\mathcal{C},\mathbf{Set}] \to \mathbf{Set}$ preserves $\mathbb{D}$-limits of representables.

Soundness is a very particular condition. All of the doctrines I mentioned above are sound. Some interesting unsound doctrines include:

The doctrine $\lambda\mathbf{-PR}$ of discrete categories with $\lt \lambda$ morphisms. The unsoundness here is disappointing if one is interested in doing algebra with infinitary operations. Some of the theory can be extended, but the notion of $\lambda$-sifted colimit used has to be more subtle.

The doctrine $\mathbf{PB}$ of pullbacks. The category of small categories, for example, is sketchable by a pullback sketch, so it’s too bad that this doctrine is unsound.

The doctrine $\mathbf{PB} \cup \mathbf{TERM}$ of pullbacks and terminal objects. This is striking given that all finite limits can be constructed out of pullbacks and terminal objects, and $\mathbf{FIN}$ is sound.

### Conclusion

Adámek, Borceux, Lack, and Rosický present their theory as a *classification* of accessible categories, emphasizing the perspective it brings on the menagerie of accessible categories out there. Most importantly, the theory puts $\mathbf{FINPR}$-accessibility on the same formal footing as the theory of $\lambda$-accessibility– it’s not clear that there are really any other important sound doctrines out there. Consider this a challenge to find more sound doctrines! At least as great a contribution is provided by the perspective that the $\mathbb{D}$-relative approach brings in organizing the general theory of $\mathbb{D}$-accessibility, allowing us to see which parts are formal and where special facts enter in.

The theory of limits and colimits that commute in $\mathbf{Set}$ has seen interesting developments recently, some reported on this blog. It’s worth mentioning that according to Marie Bjerrum, who studies these things, the better concept to work with may actually representable $\mathbb{D}$-filteredness rather than $\mathbb{D}$-filteredness. It would be interesting to see if variations of these concepts might allow the theory of this paper to be extended beyond sound doctrines.

There are also other generalizations to consider. Besides changing our doctrine $\mathbb{D}$, there is a theory of enriched accessibility, and we can also talk about taking “models” in categories other than $\mathbf{Set}$. All three directions of generalization are considered by Lack and Rosický in Notions of Lawvere Theory, but there is more to do. I believe there is also a theory of accessibility of quasicategories, and perhaps also in other contexts. All of these areas could potentially benefit from the perspective brought by relativizing to a limit doctrine.

Finally, there is more in this paper that I haven’t discussed, including a theory of $\mathbb{D}$-multipresentable categories and an interesting distributive law between the 2-monad $\mathbb{D}\mathbf{-Ind}$ and the free completion 2-monad. These might be worth discussing in the comments.

## Re: Classifying by Generalizing: The Theory of Accessibility Relative to a Limit Doctrine

Another way to think about $\mathbb{D}$-flatness is that it is just a

weightedversion of $\mathbb{D}$-filteredness.After all, in Thm 2.4 (quoted above), criterion (2) for flatness of $F: \mathcal{A}^\mathrm{op} \to \mathbf{Set}$ says that the left Kan extension along the Yoneda embedding, $\mathrm{Lan}_Y F: [\mathcal{A}, \mathbf{Set}] \to \mathbf{Set}$, is $\mathbb{D}$-continuous. But $\mathrm{Lan}_Y F$ has another name: It is just the functor $F * (-)$ which takes the $F$-weighted colimit. So criterion (2) for $\mathbb{D}$-flatness really says that $F$, considered as a colimit weight, is $\mathbb{D}$-filtered in a weighted sense.

Similarly, criterion (1) for $\mathbb{D}$-flatness of $F$ (which says that $\mathrm{Lan}_Y F$ preserves $\mathbb{D}$-limits of representables) says that $F *(-)$ preserves $\mathbb{D}$-limits of representables, i.e. that $F$ is representably $\mathbb{D}$-filtered in a weighted sense. So it would make sense to call an $F$ satisfying criterion (1) “representably $\mathbb{D}$-flat”.

In fact, when you look at the proof of the theorem, you see that it actually shows that $F$ is representably $\mathbb{D}$-flat if and only if $\mathsf{Elts}(F)^\mathrm{op}$ is representably $\mathbb{D}$-filtered – even if $\mathbb{D}$ is unsound. This is a very interesting fact. Certainly every $F$-weighted colimit can be expressed as a conical colimit over $\mathsf{Elts}(F)^\mathrm{op}$, but in the converse direction, an $\mathsf{Elts}(F)^\mathrm{op}$-diagram only corresponds to an $F$-weighted diagram if it factors through the discrete opfibration $\mathsf{Elts}(F)^\mathrm{op} \to \mathcal{A}$.