nLab
walking structure

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

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

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

Remark

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

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 ‘XX’ of “the walking XX” 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,

References

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

Revised on July 24, 2014 13:53:51 by Colin Zwanziger (174.63.87.107)