# Walking structures

## Idea

Around the nLab and elsewhere, one occasionally sees an expression “the walking _____” where the blank is some mathematical concept. This is a colloquial way of referring to an archetypal model of the concept or type, and usually refers to a free or initial context in which such a type can be interpreted. The term is believed to have been introduced by James Dolan (see the reference below).

## Definition

The idea is probably easier to apprehend through examples rather than through a formal definition, but for the record:

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

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

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

###### Remark

In other words, the structured category $[X]$ equipped with its canonical type $X$ is initial among such structured categories that come equipped with such types $X$. A fancier expression is that $[X]$ ‘coclassifies’ such types: this is analogous to how a classifying space $B G$ for a topological group $G$ classifies $G$-bundles, in that every $G$-bundle $p: E \to X$ over a suitable space $X$ has a classifying map $\chi_p: X \to B G$ (unique up to homotopy) such that pulling back the canonical $G$-bundle type $\pi: E G \to B G$ along $\chi_p$ reproduces the type $p$. Only here we say ‘coclassifies’ (as for example in this comment), since here we instead “push forward” the canonical type $X$ of $[X]$ along a structured-category morphism $[X] \to C$ to obtain a given type of $C$.

Pronunciation is just as in ‘John is a walking almanac’ or ‘Eugene Levy is a walking pair of eyebrows’.

## Examples

• The interval category is the walking arrow.

• The augmented/algebraist’s simplex category is “the walking monoid” (in a monoidal category). That is to say: the simplex category is initial (in a 2-categorical sense) among monoidal categories equipped with a monoid object. Intuitively, it is the monoidal category that results if all one need be told about it is that it has a monoid object – all the morphisms of the category are obtainable from the monoid structure by applying the operations of a monoidal category, and they are subject to no further relations beyond those implied by the monoid axioms.

• Similarly, the Lawvere theory of groups can be described as “the walking group” (qua cartesian monoidal categories). This gives a good intuitive description: this Lawvere theory can be understood as the category (with finite products) that results if all one need be told about it is that it has a group object; the rest of the structure of this category can be deduced from this one fact.

The last two examples indicate the need for a little care: the doctrine or type of structured category in which the ‘$X$’ of “the walking $X$” lives should either be specified or clear from context. For example, if one simply says “the walking monoid”, this means the simplex category if the surrounding context is the doctrine of monoidal categories – but means something else (the Lawvere theory of monoids) if the ambient context is the doctrine of categories with finite products, and it means the category opposite to that of finitely presentable monoids if we are in the doctrine of finitely complete categories.

If the doctrine is not specified, then a reasonable default is a ‘minimal’ doctrine in which the concept makes sense; for example, to make sense of monoids, one doesn’t need more than monoidal categories. See also microcosm principle.

Thus, more generally,

• 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 $T$-modelqua Grothendieck toposes (where the morphisms are the left-adjoint parts of geometric morphisms).

## Relation to initial objects

The walking X is, of course, not the same as the initial X.

Consider for example the case when X is a pointed monoid (a monoid equipped with an element). The initial pointed monoid (in $Set$) is the natural numbers equipped with $1\in \mathbb{N}$. Whereas the walking pointed monoid (qua monoidal category, say) is a monoidal category $C_M$ containing a monoid object $M\in C_M$ and an “element$e:I\to M$, where $I$ is the unit object of $C_M$. They have different types and different universal properties: $\mathbb{N}$ has a universal property mapping into other pointed monoids in $Set$, while $C_M$ has a universal property mapping into other monoidal categories equipped with pointed monoids.

Nevertheless, the first sits inside the second! Specifically, for any monoidal category $C$, the “underlying set” functor $C(I,-):C\to Set$ is lax monoidal and hence carries monoid objects to monoid objects, and in the case of the walking pointed monoid we have $C_M(I,M) \cong \mathbb{N}$.

This is true rather generally: the initial X is the underlying X of the walking X. One general theorem of this sort is the following:

###### Theorem

Let $K$ be a 2-category containing an object $S$, and suppose that:

1. The domain projection $K\sslash S \to K$ from the lax slice 2-category has a section. Explicitly, for every object $X$ we have a map $s_X : X\to S$ and for every morphism $f:X\to Y$ we have a 2-cell $\sigma_f : s_X \to s_Y f$, such that for every 2-cell $\alpha :f\to g$ we have $\sigma_g . s_Y\alpha = \sigma_f$, and these vary functorially.
2. We have $s_S \cong 1_S$, and for any $X$ the composite $s_X \xrightarrow{\sigma_{s_X}} s_S s_X \cong s_X$ is the identity.

Then for any $X$, the morphism $s_X:X\to S$ is the initial object of the hom-category $K(X,S)$.

###### Proof

We will use the characterization of initial objects via cones over the identity. Thus, we must construct a natural transformation from the constant functor $\Delta_{s_X} : K(X,S) \to K(X,S)$ to the identity, which is the identity at $s_X$. However, given any $f:X\to S$, we have the 2-cell $s_X \xrightarrow{\sigma_f} s_S f \cong f$, and the assumption $\sigma_g . s_Y\alpha = \sigma_f$ makes this a natural transformation; and the final assumption says exactly that this is the identity at $s_X$.

To see how this theorem implies that the initial X is the underlying X of the walking X, consider again the case of pointed monoids. Let $K$ be the 2-category of monoidal categories and lax monoidal functors, let $S=Set$, and let $s_C : C \to Set$ be $C(I,_)$. The hypotheses are easy to verify; thus the theorem tells us that $C(I,-)$ is the initial functor $C\to Set$ for any $C$.

Now take $C$ to be the walking pointed monoid $C_M$ above. Then its universal property tells us that functors $F:C_M\to Set$ are equivalent to pointed monoids $F(M)$ in $Set$; so we see that $C(I,M)$ is the initial pointed monoid in $Set$, i.e. $\mathbb{N}$.

A similar argument applies whenever we have a “$Set$-like” object of a 2-category with “underlying set” morphisms to it. For instance, in the 2-category of monoidal double categories? we can take $S$ to be the double category Span.

## References

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

Revised on July 1, 2017 06:06:25 by Mike Shulman (76.167.222.204)