nLab
interval category

Definition

Interval category

The interval category – often denoted $2$ or $I$ or $\Delta [1]$ – is the category with two objects and precisely one nontrivial morphism connecting them:

$$I=\{0\to 1\}.$$I = \left\{ 0 \to 1 \right\} \,.

The interval category serves as a combinatorial model for the directed interval. It is the directed canonical interval object in Cat. It is also called the walking arrow. It might also be called the “arrow category” although that term is also used for a category of functors out of $2$.

The notation $2$ comes from the fact that the interval category is also the ordinal number $2$ regarded as a poset, regarded as a category.

It also appears as

• the 1-simplex;

• the first oriental;

• the 1-globe.

Since every category is also an (n,r)-category for $n,r\ge 1$, we may regard $I$ also as some $(n,r)$-category. For instance regarded as a (∞,1)-category and modeled as a quasi-category, the interval category is the simplicial set $\Delta [1]$.

Interval groupoid

The interval groupoid is a combinatorial model for the undirected interval.

It is the free groupoid on the interval category, where the morphism $0\to 1$ is an isomorphism. Accordingly the interval groupoid has a second nontrivial morphism, the inverse $1\to 0$.

This is the undirected interval object in Cat and in Grpd.

Applications

The interval category is one of those diagram category that are not terribly interesting in themselves, but that serve an important role in category theory as a whole.

For instance a natural transformation $\eta :F\Rightarrow G$ between two functors $F,G:C\to D$ is precisely the same as a strictly commuting diagram

$$\begin{array}{c}C\times \{0\}\\ \downarrow & {\searrow }^F\\ C\times I& \stackrel{\eta }{\to }& D\\ \uparrow & {\nearrow }_G\\ C\times \{1\}\end{array}$$\array{ C \times \{0\} \\ \downarrow & \searrow^{\mathrlap{F}} \\ C \times I &\stackrel{\eta}{\to}& D \\ \uparrow & \nearrow_{\mathrlap{G}} \\ C \times \{1\} }

in Cat, where on the left we have the cartesian product of $C$ with $I$.

Accordingly, for $I_{\mathrm{iso}}$ the interval groupoid, a natural isomorphism $\eta :F\Rightarrow G$ is the same as a diagram

$$\begin{array}{c}C\times \{0\}\\ \downarrow & {\searrow }^F\\ C\times I_{\mathrm{iso}}& \stackrel{\eta }{\to }& D\\ \uparrow & {\nearrow }_G\\ C\times \{1\}\end{array}.$$\array{ C \times \{0\} \\ \downarrow & \searrow^{\mathrlap{F}} \\ C \times I_{iso} &\stackrel{\eta}{\to}& D \\ \uparrow & \nearrow_{\mathrlap{G}} \\ C \times \{1\} } \,.

This is a left homotopy in $\mathrm{Cat}$.

Dually, forming the functor category

$$\mathrm{Arr}(D):=[I,D]$$Arr(D) := [I,D]

from the interval category produces the arrow category of $D$, and a natural transformation $\eta$ is also the same as a diagram

$$\begin{array}{ccc}& & D\times \{0\}\\ & {}^F\nearrow & \uparrow \\ C& \stackrel{\eta }{\to }& [I,D]\\ & {}_G\searrow & \downarrow \\ & & D\times \{1\}\end{array}.$$\array{ && D \times \{0\} \\ & {}^{\mathllap{F}}\nearrow & \uparrow \\ C &\stackrel{\eta}{\to}& [I,D] \\ & {}_{\mathllap{G}}\searrow & \downarrow \\ && D \times \{1\} } \,.

With $I$ replaced by $I_{\mathrm{iso}}$ this is again a natural isomorphism, now represented as a right homotopy in Cat.

The analogous statements are true in higher category theory.