Contents

category theory

# Contents

## Idea

A direct category is a category in which the morphisms only go in “one direction.”

A direct category can be thought of as a category of geometric shapes for higher structures which includes only the “inclusions of faces,” not any “degeneracy” maps. (The more general notion of Reedy category can also include degeneracies.) The objects of a direct category admit no nontrivial automorphisms, but the notion can also be generalized to allow for such automorphisms.

Alternately, a direct category can be thought of as a combinatorial representation of a single geometric shape. In this case, each object is a “face” and each morphism is the inclusion of a lower-dimensional face in a higher-dimensional one. If $D$ is a direct category regarded as a category of geometric shapes, then each $d\in D$ can be represented by the slice category $D/d$, a direct category regarded as a single geometric shape. (In general, however, $D/d$ may admit more automorphisms than $d$, so it may need to be equipped with a “labeling” or “orientation” to truly recapture the shape $d$.)

Finally, a direct category can also be thought of as a categorification of the notion of well-founded relation from posets to categories.

## Definition

### Standard definition

A category $D$ is a direct category if the following equivalent conditions are satisfied.

• $D$ contains no infinite descending chains of nonidentity morphisms $\cdots \to\cdot \to\cdot\to\cdot$ (including cycles of length $\gt 0$).
• The relation $a\prec b$ on $ob(D)$ defined by “there exists a nonidentity morphism from $a$ to $b$” is well-founded.
• There exists a function $d\colon ob(D)\to Ord$, where $Ord$ is the class of ordinals, such that every nonidentity morphism of $D$ raises the degree.
• There exists an identity-reflecting functor (that is, it maps a morphism to an identity if and only if the morphism is itself an identity) $d : D\to \mathbf{Ord}$, where $\mathbf{Ord}$ is the large poset of ordinals viewed as a category.
• $D$ is a Reedy category (in particular an elegant Reedy category) in which $D_-$ consists only of identity maps (or equivalently $D_+$ is all of $D$).

In particular, a poset is a direct category just when its strict order relation $\lt$ is well-founded. Thus, direct categories can be seen as a categorification of well-founded relations.

If $D^{op}$ is a direct category, then we say that $D$ is an inverse category.

### Direct versus one-way categories

Of course, a direct category can have no nonidentity endomorphisms, since any such endomorphism would be a cycle of length 1. A category with this property is sometimes called a one-way category.

A direct category is also necessarily skeletal, since any nonidentity isomorphism and its inverse would form a cycle of length 2. The notion of generalized direct category, below, relaxes this requirement.

Conversely, a skeletal one-way category can have no cycles of nonidentity morphisms of any finite length, since by the one-way property all morphisms in such a cycle would be isomorphisms, hence endomorphisms by skeletality, and hence identities by the one-way property. Since any infinite chain in a finite category contains a cycle, we see that

• A finite category is a direct category if and only if it is one-way and skeletal.

Since this condition is self-dual, we also see that

• A finite category is direct if and only if it is inverse.

An infinite category can be one-way and skeletal without being direct or inverse, such as the poset $\mathbb{Z}$ with its usual ordering. However, if it additionally has finite fan-out, meaning that for each object $x$ there are altogether only finitely many morphisms with source $x$ (and arbitrary target), then it must be an inverse category, for any infinite non-cyclic chain would induce infinitely many distinct morphisms out of any of its objects by composition. In FOLDS, skeletal one-way categories with finite fan-out are called simple categories and used as signatures; thus

• Any simple category (in the sense of FOLDS) is an inverse category.

However, a category can be inverse without having finite fan-out. Let $S$ be an infinite set and consider the poset $S \cup \{\infty\}$, with $S$ having the discrete ordering (no nonidentity arrows) and $\infty$ being less than every element of $S$. This does not have finite fan-out, since $\infty$ is the source of infinitely many distinct arrows, but it is inverse, since there are clearly no chains of nonidentity arrows of length $\gt 1$.

### Allowing automorphisms

Some categories of geometric shapes, such as the tree category $\Omega$ and the cycle category $\Lambda$, include automorphisms of their objects. By analogy with the notion of generalized Reedy category, we can define $D$ to be a generalized direct category by replacing “identity” with “isomorphism” in the above definition. Thereby we obtain the following equivalent conditions for $D$ to be a generalized direct category.

• $D$ contains no infinite descending chains of noninvertible morphisms $\cdots \to\cdot \to\cdot\to\cdot$.
• The relation $a\prec b$ on $ob(D)$ defined by “there exists a noninvertible morphism from $a$ to $b$” is well-founded.
• There exists a function $d\colon ob(D)\to Ord$, where $Ord$ is the class of ordinals, such that every noninvertible morphism of $D$ raises the degree.
• There exists an isomorphism-reflecting functor (that is, it maps a morphism to an isomorphism if and only if the morphism is itself an isomorphism) $d : D\to \mathbf{Ord}$, where $\mathbf{Ord}$ is the large poset of ordinals viewed as a category.
• $D$ is a generalized Reedy category in which $D_-$ consists only of isomorphisms (or equivalently $D_+$ is all of $D$).

Of course, $\Omega$ and $\Lambda$ are not generalized direct categories themselves, since they have degree-lowering degeneracies as well, but their full subcategories of coface maps are generalized-direct. More generally:

• If $R$ is any generalized Reedy category, then $R_+$ is a generalized direct category.

### Direct categories versus finite simplicial sets

If a direct category has finitely many objects then its nerve is a finite simplicial set. Conversely, if a finite simplicial set is the nerve of a category then the category is a direct category with finitely many objects.

## Properties

### Model structures

Every direct category (and every inverse category) is in particular a Reedy category, in fact an elegant Reedy category. Therefore whenever $M$ is a model category there is a Reedy model structure on $M^D$. In the case of direct and inverse categories, these model structures are even easier to describe, since either the latching or the matching objects are degenerate.

## Examples

### Inside the simplex category

The wide subcategory $\Delta_+$ of the simplex category on the injective maps is direct. That is, the category with objects being the totally ordered sets $\{0\}, \{0, 1\}, \{0, 1, 2\}...$, and the morphisms being injective order-preserving maps. Its presheaves are semi-simplicial objects/semi-simplicial sets as opposed to simplicial objects/simplicial sets.

### The direct category of corollas

We now define a “universal” generalized direct category which contains “all” geometric shapes for higher structures. This is based on (Borisov).

A functor $f\colon D\to E$ between generalized direct categories is called a dependency if it is equivalent to the inclusion of a sieve, or equivalently if it is a fully faithful functor and a discrete fibration in the generalized sense of Street. If $D$ and $E$ are skeletal (as they must be if they are non-generalized direct categories), then any dependency must be isomorphic to the inclusion of a sieve. One also usually works only with skeletal generalized direct categories, although the definition does not require it.

If we regard direct categories as a categorification of well-founded relations, then dependencies are a categorification of injective simulations, i.e. the inclusions of initial segments.

Define a corolla to be a generalized direct category with a weakly terminal object; we call this object the vertex of the corolla. If a generalized direct category is finite and skeletal, then it is a corolla if and only if it has a unique object which is not the source of any noninvertible morphism. Corollas are the direct categories which it is most natural to regard as single geometric shapes; other direct categories are more like “pasting diagrams” of geometric shapes.

Let $Corolla$ be the category of small skeletal corollas and dependencies.

###### Theorem

$Corolla$ is a (large) generalized direct category.

###### Proof

Suppose that $\cdots \overset{d_2}{\to} C_2 \overset{d_1}{\to} C_1 \overset{d_0}{\to} C_0$ is a descending chain of noninvertible dependencies. Any dependency whose target is a corolla and whose image contains the vertex must be an equivalence, and hence an isomorphism if the corollas are skeletal; thus none of the dependencies $C_{n+1} \to C_{n}$ can have the vertex in their image. Let $c_n\in C_0$ be the image of the vertex $\star_n \in C_n$ under the composite $d_0\circ \dots \circ d_{n-1}$. Then we have a noninvertible morphism $c_{n+1} \to c_n$ for each $n$, arising from some map $d_n(\star_{n+1}) \to \star_n$ which exists since $\star_n$ is weakly terminal. This is a contradiction, since $C_0$ is a generalized direct category.

Of course, the same is true for any subcategory of $Corolla$. In particular, it is very natural to consider only the category $FinCorolla$ of finite corollas, which is moreover essentially small (though not finite).

We next observe that $Corolla$ is the “universal” generalized direct category in a certain sense. Let $D$ be any generalized direct category; then the slice category $D/d$ is a corolla for any $d\in D$. Moreover, if $f\colon d\to d'$ is a morphism in $D$ which is monic, then the “composition” functor $\Sigma_f\colon D/d \to D/d'$ is a dependency. Thus, if every morphism in $D$ is monic, we have a functor $ext_D\colon D \to Corolla$ with $ext_D(d)= D/d$ and $ext_D(f)=\Sigma_f$.

Now if $f,g\colon d\to d'$ are parallel morphisms and $\Sigma_f=\Sigma_g$, then in particular $f = \Sigma_f(id_d) = \Sigma_g(id_d) = g$; thus $ext_D$ is faithful. Therefore, if $D$ is a direct category in which all morphisms are monic, and in which $D/d \cong D/d'$ implies $d=d'$ (which includes many examples), then $D$ is equivalent to a subcategory of $Corolla$.

This subcategory of $Corolla$ is usually not full, however. In particular, for $d\in D$ the corolla $D/d$ will generally admit more automorphisms than $d$ has in $D$. For instance, if $D$ is a non-generalized direct category, then $d$ has no nontrivial automorphisms, whereas $D/d$ generally will. In particular, if the objects of $D$ have any sort of “orientation” or “labeling,” then this information is forgotten by the functor $ext_D$.