nLab
walking structure

Definition

If X is a type of structure that can be defined in a category, higher category, or category with some sort of structure, then the walking X is an informal term for the free category (resp. higher category, category with suitable structure) containing an X.

More precisely, if $\mathrm{StructCat}$ denotes some (higher) category of categories with an appropriate type of structure, then the walking X is an object $[X]\in \mathrm{StructCat}$ together with a natural equivalence

$$\mathrm{StructCat}([X],C)\simeq \{\mathrm{Xs}\mathrm{in}C\}$$StructCat([X],C) \simeq \{Xs \; in \; C\}

between the hom-set/category/space from $[X]$ to $C$, for any $C\in \mathrm{StructCat}$, and the set/category/space of all Xs in $C$.

Examples

• The interval category is the walking arrow.

• The augmented/algebraist’s simplex category is the walking monoid (in a monoidal category).

• The syntactic category of a theory $T$ in some doctrine $D$ is the “walking $T$-model” (in a $D$-category). In particular, the classifying topos of a geometric theory $T$ is the “walking $T$-model” qua Grothendieck topos (where the morphisms are the left-adjoint parts of geometric morphisms).

References

• A Café post about walking objects (among other things), including a comment that explains the terminology.