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 is a direct category regarded as a category of geometric shapes, then each can be represented by the slice category , a direct category regarded as a single geometric shape. (In general, however, may admit more automorphisms than , so it may need to be equipped with a “labeling” or “orientation” to truly recapture the shape .)
A category is a direct category if the following equivalent conditions are satisfied.
In particular, a poset is a direct category just when its strict order relation is well-founded. Thus, direct categories can be seen as a categorification of well-founded relations.
If is a direct category, then we say that is an inverse category.
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
Since this condition is self-dual, we also see that
An infinite category can be one-way and skeletal without being direct or inverse, such as the poset with its usual ordering. However, if it additionally has finite fan-out, meaning that for each object there are altogether only finitely many morphisms with source (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
However, a category can be inverse without having finite fan-out. Let be an infinite set and consider the poset , with having the discrete ordering (no nonidentity arrows) and being less than every element of . This does not have finite fan-out, since is the source of infinitely many distinct arrows, but it is inverse, since there are clearly no chains of nonidentity arrows of length .
Some categories of geometric shapes, such as the tree category and the cycle category , include automorphisms of their objects. By analogy with the notion of generalized Reedy category, we can define to be a generalized direct category by replacing “identity” with “isomorphism” in the above definition. Thereby we obtain the following equivalent conditions for to be a generalized direct category.
Of course, and 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:
Every direct category (and every inverse category) is in particular a Reedy category, in fact an elegant Reedy category. Therefore whenever is a model category there is a Reedy model structure on . 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.
The wide subcategory of the simplex category on the injective map (the co-face maps) is direct. Its presheaves are semi-simplicial objects/semi-simplicial sets as opposed to simplicial objects/simplicial sets.
A functor 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 and 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 be the category of small skeletal corollas and dependencies.
is a (large) generalized direct category.
Suppose that 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 can have the vertex in their image. Let be the image of the vertex under the composite . Then we have a noninvertible morphism for each , arising from some map which exists since is weakly terminal. This is a contradiction, since is a generalized direct category.
Of course, the same is true for any subcategory of . In particular, it is very natural to consider only the category of finite corollas, which is moreover essentially small (though not finite).
We next observe that is the “universal” generalized direct category in a certain sense. Let be any generalized direct category; then the slice category is a corolla for any . Moreover, if is a morphism in which is monic, then the “composition” functor is a dependency. Thus, if every morphism in is monic, we have a functor with and .
Now if are parallel morphisms and , then in particular ; thus is faithful. Therefore, if is a direct category in which all morphisms are monic, and in which implies (which includes many examples), then is equivalent to a subcategory of .
This subcategory of is usually not full, however. In particular, for the corolla will generally admit more automorphisms than has in . For instance, if is a non-generalized direct category, then has no nontrivial automorphisms, whereas generally will. In particular, if the objects of have any sort of “orientation” or “labeling,” then this information is forgotten by the functor .