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category theory

topos theory

Contents

Idea

The dense topology is a Grothendieck topology $J_d$ on a small category $\mathcal{C}$ whose sieves generalize the idea of a ‘downward dense’ poset. The corresponding sheaf topos $Sh(\mathcal{C},J_d)$ yields the double negation subtopos of the presheaf topos on $\mathcal{C}$.

The dense topology is important for sheaf-theoretic approaches to forcing in set theory (cf. continuum hypothesis).

There is also a closely related but more general concept of a dense Lawvere-Tierney topology which is discussed at dense subtopos.

Definition

Let $\mathcal{C}$ be a category. The dense topology $J_d$ is the Grothendieck topology with collections of sieves $J_d(Y)$ of the form:

A sieve $S$ is in $J_d(Y)$ iff for all $f:X\to Y$ there exists a $g:Z\to X$ such that $f\circ g:Z\to Y$ is in $S$.

Properties

Recall that a category $\mathcal{C}$ satisfies the Ore condition if every diagram $X\rightarrow W\leftarrow Y$ can be completed to a commutative square. In this case the dense topology has a simpler description as the collection of all nonempty sieves and is called the atomic topology $J_{at}$ on $\mathcal{C}$ :

Proposition

Let $\mathcal{C}$ be a category satisfying the Ore condition. Then the dense topology $J_d$ coincides with the atomic topology $J_{at}$.

For the (easy) argument see at atomic site. One direction relies on the following straight forward

Observation

If $S\in J_d$ then $S\neq \emptyset$ . $\qed$

Note that the coincidence with the atomic topology on categories satisfying the Ore condition affords for the sheaves of $J_d$ the following simple description in such cases:

Proposition

Let $(\mathcal{C}, J_{d})$ be a site such that $\mathcal{C}$ satisfies the Ore condition. A presheaf $P\in Set^{\mathcal{C}^{op}}$ is a sheaf for $J_{d}$ iff for any morphism $f:D\to C$ and any $y\in P(D)$ , if $P(g)(y)=P(h)(y)$ for all diagrams

$E\overset{g}{\underset{h}{\rightrightarrows}} D\overset{f}{\to} C$

with $f\cdot g=f\cdot h$ , then $y=P(f)(x)$ for a unique $x\in P(C)$.

For the proof see Mac Lane-Moerdijk (1994, pp.126f).

Remark

It is possible to define the atomic topology $J_{at}$ on arbitrary categories $\mathcal{C}$ as the smallest topology containing all non-empty sieves. Then observation implies that $J_d\subseteq J_{at}$, and accordingly, $Sh(\mathcal{C},J_{at})\subseteq Sh(\mathcal{C},J_d)$ . But $J_d=J_{at}$ precisely if $\mathcal{C}$ satisfies the Ore condition (for details see at atomic site).

The next result warrants the importance of the dense topology:

Proposition

For every small category $\mathcal{C}$, the Lawvere-Tierney topology on the presheaf topos $Set^{\mathcal{C}^{op}}$ corresponding to the dense topology on $\mathcal{C}$ is the double negation topology $\neg\neg$ on $Set^{\mathcal{C}^{op}}$ . In other words, $Sh(\mathcal{C},J_d)\simeq Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ .

This appears as (MacLaneMoerdijk, corollary VI 5).

In particular, the Lawvere-Tierney topology corresponding to the dense topology is dense as a Lawvere-Tierney topology!

Reference

Last revised on June 30, 2019 at 10:23:30. See the history of this page for a list of all contributions to it.