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dense topology

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Idea

The dense topology is a Grothendieck topology J dJ_d on a small category 𝒞\mathcal{C} whose sieves generalize the idea of a ‘downward dense’ poset. The corresponding sheaf topos Sh(𝒞,J d)Sh(\mathcal{C},J_d) yields the double negation subtopos of the presheaf topos on 𝒞\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 𝒞\mathcal{C} be a category. The dense topology J dJ_d is the Grothendieck topology with collections of sieves J d(Y)J_d(Y) of the form:

A sieve SS is in J d(Y)J_d(Y) iff for all f:XYf:X\to Y there exists a g:ZXg:Z\to X such that fg:ZYf\circ g:Z\to Y is in SS.

Properties

Recall that a category 𝒞\mathcal{C} satisfies the Ore condition if every diagram XWYX\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 atJ_{at} on 𝒞\mathcal{C} :

Proposition

Let 𝒞\mathcal{C} be a category satisfying the Ore condition. Then the dense topology J dJ_d coincides with the atomic topology J atJ_{at}.

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

Observation

If SJ dS\in J_d then SS\neq \emptyset . \qed

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

Proposition

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

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

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

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

Remark

It is possible to define the atomic topology J atJ_{at} on arbitrary categories 𝒞\mathcal{C} as the smallest topology containing all non-empty sieves. Then observation implies that J dJ atJ_d\subseteq J_{at}, and accordingly, Sh(𝒞,J at)Sh(𝒞,J d)Sh(\mathcal{C},J_{at})\subseteq Sh(\mathcal{C},J_d) . But J d=J atJ_d=J_{at} precisely if 𝒞\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 𝒞\mathcal{C}, the Lawvere-Tierney topology on the presheaf topos Set 𝒞 opSet^{\mathcal{C}^{op}} corresponding to the dense topology on 𝒞\mathcal{C} is the double negation topology ¬¬\neg\neg on Set 𝒞 opSet^{\mathcal{C}^{op}} . In other words, Sh(𝒞,J d)Sh ¬¬(Set 𝒞 op)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.