# Idea

In the sense of space and quantity a space is, quite generally, a presheaf on some category of test objects.

An $\infty$-space is some ∞-categorification of this.

With some category $Sp \subset PSh(C)$ of spaces (1-spaces, that is) fixed, an $\infty$-space may for instance be modeled as an ∞-groupoid internal to $Sp$.

If $Sp$ is for instance a category of smooth test objects, such as Diff or CartSp, then we have

Generally, the notion of ∞-groupoid internal to $PSh(C)$ is naturally interpreted as a simplicial presheaf. Indeed, when equipped with the model structure on simplicial presheaves, simplicial presheaves are good models for ∞-stack (∞,1)-toposes.

The $\infty$-stack terminology is possibly more familiar than that of $\infty$-spaces. To some extent the usefulness of both terminologies depends on whether $Sp$ is a petit topos of presheaves of open subsets in some fixed topological space or a gros topos of presheaves on general test spaces, such as on (an essentially small) version of Top itself.

In the former case of a petit topos, the $\infty$-stack terminology may be more suggestive. In the latter case of a gros topos the $\infty$-space perspective typically conveys the right intuition for the objects under consideration much better.

# Warning on terminology

The idea, that “space” is fundamentally to be interpreted in the sense of space and quantity is only partially compatible with wide-spread use of the word “space” to mean concretely topological space.

In fact, in the context of “$\infty$-space” there is a subtle confusion of terms (possible) here, which deserves to be carefully sorted out:

In higher category theory topological spaces $X$ are often, via the homotopy hypothesis, effectively identified with their fundamental ∞-groupoids $\Pi(X)$. The ∞-groupoid $\Pi(X)$ – as an $\infty$-groupoid internal to Set – is really a discrete $\infty$-space in the above sense, namely an $\infty$-groupoid in the category of presheaves on the point.

# $\infty$-Quantities

Following the duality of space and quantity, the concept dual to $\infty$-space is ∞-quantity. See there for more details.

# Referemces

One place where the conceptual usefulness of interpreting categories internal to smooth spaces as smooth 2-spaces (and de-emphasizing conceptually their realization as simplicial sheaves) has been particularly amplified is the work

Revised on December 20, 2011 15:16:24 by David Corfield (86.159.145.83)