nLab undirected object

Idea

In a category with interval object one has for every object $X$ a notion of paths in $X$. (Indeed, the very definition of category with interval object is meant to guarantee that for every object there is its fundamental category in the Trimblean sense).

The idea is that if for all these paths in $X$ there is a reverse path, then the object $X$ is undirected or groupoidal. Otherwise it is strictly directed (not to be confused with a directed object).

Definition (tentative)

Let $V$ be a category with an interval object $I$. Recall that there is for every object $X$ of $V$ a canonical morphism $X_0 \to [I,X]$ which embeds the points of $X$, $X_0 := [pt,X]$, as the constant paths into the “object of directed paths” $[I,X]$.

An object $X$ in $V$ is called undirected or an undirected object with respect to $I$, – or $I$-undirected – if this morphism is a weak equivalence:

$$(X is undirected) \Leftrightarrow ([\pt,X] \stackrel{\simeq}{\to} [I,X]) \,;$$

otherwise it is called strictly directed or a strictly directed object with respect to $I$, or strictly $I$-directed.

Examples

• $V =$ Top with its standard model category structure, and $I = [0,1]$ the standard interval. Then all objects are undirected since the interval is contractible.

• $V = \omega Cat$, strict omega-categories, equipped with the folk model structure and $I = \{a \to b\}$ the 1-globe, the first oriental. Then

  • strict $\omega$-omega-groupoids are undirected;
  • in particular the interval $I$ itself is strictly directed, as there is no weak equivalence from $I_0 = I$ to $[I,I] = \{a \to b \to c\}$;
  • in general, the undirected objects should be precisely those for which every $j$-morphism is an $\omega$-equivalence. These should consist of those $\infty$-infinity-groupoids that are strict as $\infty$-categories, or those objects that are weakly equivalent to a strict $\omega$-groupoid. For this purpose it is essential that the internal-hom for the Crans-Gray tensor product on strict $\omega$-categories contains lax transformations, modifications, and so on; if it only contained pseudo natural transformations, then the undirected objects would just be those for which every 1-morphism is an equivalence.

• Once we have a closed monoidal homotopical structure on the category $dTop$ of directed topological spaces, it should be true that in $V = dTop$ with $I = I_d$ the standard directed interval, an object $(X, d X)$ is undirected precisely if it contains no nontrivial directed path.

Remarks

Whether or not an object is undirected depends on the choice of interval object.

• For instance the terminal object $pt$ itself is always a an interval object, $I = pt$, but in a trivial way. Every object is $pt$-undirected.

• A bit more generally, the interval object may be not equal to the terminal object, but still be weakly equivalent to it. For instance the standard (undirected) topological interval $[0,1]$ is not equal to but is weakly equivalent to the point in the standard model structure on topological spaces. This implies in particular that with respect to the standard undirected topological interval even a directed topological space should undirected. (Well, we still need to specify the closed structure on $dTop$ to make this true…) It’s only the standard directed topological interval which can detect the directedness of a strictly directed topological space.

• Analogous statements are true in the example of strict omega-categories, where the analogue of the standard directed topological interval is the 1-globe $I_{dir} = \{a \to b\}$, while the example of the standard undirected topological interval is the groupoid version of this, $I_{inv} = \{a \stackrel{\simeq}{\to} b\}$, where the morphism from $a$ to $b$ is an isomorphism. As opposed to $I_{dir}$ this $I_{inv}$ is weakly equivalent to the terminal category $pt = \{\bullet\}$ (the 0-globe) (with respect to the folk model structure). It should be true that every strict $\omega$-category is $I_{inv}$-undirected, even if it is strictly $I_{dir}$-directed.