nLab category with an atlas

Idea

Background comments

Despite the impression one may get from standard texts, the notion of a category (in category theory) exists in (at least) two distinct versions, which, while closely related, are crucially different in a way that matters when paying attention to fine-print. Namely, categories are traditionally introduced as being mathematical structures consisting of a class of objects, etc. (“strict categories”, see below) but in further discussion these are eventually often regarded as being themselves the objects of the 2-category $CAT$ of categories, functors and natural transformations between these. While this change of perspective may superficially seem to only increase the level of sophistication of the discussion, it actually loses information: By default of 2-category theory, the objects of a 2-category like CAT are identifiable only up to 2-categorical equivalence – which here means: up to equivalence of categories – but under this relation the notion of “class of objects” of a category is not fixed (not fixed up to isomorphism, that is) and hence makes no invariant sense: Regarding a category as an object of CAT (and nothing more), means to forget its precise class of objects.

For example, when assuming the relevant axiom of choice, then the codiscrete category on any set $S$ (whose class of objects is $S$) is equivalent to the terminal category (whose class of objects is the singleton set). From the point of view of the “principle of equivalence” appropriate within CAT, all these categories are effectively indistinguishable, and yet one finds a strict representative in this equivalence class with any imaginable class of objects.

In order to bring out this distinction, some authors say “strict category” for the notion of “category” where a class of objects is fixed. While this specific terminology is not common, the notion itself is tacitly ubiquituous: For example, when people say that the simplicial nerve-construction fully embeds small categories into simplicial sets (namely as those which satisfy the Segal conditions) then they are implicitly referring to such strict categories: Their fixed set of objects constitutes the degree-0 component of their nerve simplicial set.

The point of the following definition of “category with atlas” is to bring out the notion of “strict category” in a way that is itself intrinsic to 2-category theory – or rather: internal to 2-topos theory.

Idea of the definition

By a category with an atlas we shall mean a category $\mathcal{C}$ which is equipped with an essentially surjective functor

(1)$$C_0 \twoheadrightarrow \mathcal{C}$$

from a set (or class) $C_0$ regarded as a (large) discrete category. This is also known as a flagged category (in which case one would allow the domain $C_0$ to be itself a groupoid, but one could agree to do the same here).

This is an “invariant” or “intrinsic” way of speaking (namely internal to the 2-topos CAT) about categories which are equipped with a notion of a fixed class of objects. Another terminology for this is “strict categories”, but saying “category with atlas” naturally connects to the well-established notion of atlas in the context of stacks, maybe most commonly used in the literature on topological stacks:

An atlas for a topological stack $\mathscr{X}$ is classically taken to be a topological space $X_0$ and an effective epimorphism $X_0 \twoheadrightarrow \mathscr{X}$. Given this, its homotopy-Cech nerve $X_0 \times_{\mathscr{X}} X_0 \rightrightarrows X_0$ is known as the corresponding topological groupoid, whose space of objects is nothing but $X_0$.

By the Giraud-Rezk-Lurie theorem, this state of affairs holds in any (2,1)-topos of stacks: The (“strict”!) groupoid objects in the underlying 1-topos are equivalently the stacks equipped with an atlas, namely equivalently objects in the (2,1)-topos equipped with an effective epimorphism out of a 0-truncated object.

If here the the underlying site is the terminal category, then “stacks” are groupoids regarded up to equivalence of groupoids (i.e. as objects of the (2,1)-category Grpd, and “stacks with atlas” are groupoids equipped with a choice of a fixed set of objects, hence “strict groupoids”. This is just the notion of “category with an atlas” restricted to categories that happen to be groupoids.

In short, we have the following dictionary between “stack theory” and category theory (which are unified in 2-topos theory):

stack theory	stack theory over the point	category theory
stack	groupoid	groupoid
stack with atlas	groupoid with atlas	strict groupoid

But it is common to understand “stacks” in the generality of category-valued stacks: 2-sheaves. Now a stack/2-sheaf over the point is just a bare category, and this way the notion of “category with an atlas” blends seamlessly into established terminology in stack theory.

Consequences of the definition

The advantage of thinking of strict categories more intrinsically as categories equipped with an atlas (1) is that it immediately implies natural definitions for a variety of related notions.

For example, the correct category of categories-with-atlas is evidently the full sub 2-category of the $(2,1)$-comma category $SET/CAT$ on the essential surjections, whose morphisms from $(C_0 \twoheadrightarrow \mathcal{C})$ to $(D_0 \twoheadrightarrow \mathcal{D})$ are squares in $Ho_{(2,1)}(CAT)$ of the form

Similarly, for a given class $C_0$ of objects, the morphisms in the analogous full sub-$(2,1)$-category of the co-slice of CAT under $C_0$ are of the form:

Such morphisms are evidently equivalent (by 2-morphisms in $CAT^{C_0/}$) to “identity-on-objects functors”.

See also

• atlas

• identity-on-objects functor

• Freyd category

• Freyd multicategory

• dagger category

References

A generalization of the notion to higher category theory is considered (under the name “flagged categories”) in

• David Ayala, John Francis, Def. 0.12 of: Flagged higher categories, in: Topology and Quantum Theory in Interaction, Contemporary Mathematics 718, AMS (2018) [arXiv:1801.08973, ISBN:978-1-4704-4941-4]