duaLab

Aufhebung

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1. Definition

A level of a Grothendieck topos is an essential subtopos. Objects in the essential images of the left and right adjoints of its restriction functor are skeletal and coskeletal at that level.

The Aufhebung of a level is the least level containing it such that every object skeletal at the original level is coskeletal at the larger level, when such a least level exists.

See §1 and Definition 2.4.1.

2. Examples

Definition 2.1 (Symmetric simplicial sets)

Let $\mathbb F$ be the category of nonempty finite sets and all functions. A symmetric simplicial set is a presheaf $\mathbb F^{\mathrm{op}}\to\mathbf{Set}$.

The levels of this topos are indexed by $-\infty,0,1,2,\ldots,\infty$. The mass of a structure is the cardinality of the underlying set of its nondegenerate part in an Eilenberg–Zilber decomposition. For a finite nonnegative level $l$, the $l$-skeleton consists of structures of mass at most $l+1$.

See Definition 2.1.1 and Propositions 2.3.1 and 2.3.4.

Theorem 2.2 (Aufhebung for symmetric simplicial sets)★

For finite levels and the bottom level, the Aufhebung $a_l$ is

$$a_{-\infty}=0,\qquad a_0=1,$$
$$a_1=2,\qquad a_2=4,$$
$$a_l=2l-1\quad(l\geq3).$$

See Theorem 3.7.1.

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4. References