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Stream: deprecated: topos theory

Topic: LT topologies and subalgebras of \Omega

Matteo Capucci (he/him) (Jan 19 2021 at 10:30):

Slightly dumb question: are Lawvere-Tierney topologies on a topos $\mathcal E$ in 1-to-1 correspondence with complete Heyting sub-algebras of $\Omega$?
The idea is that given $j: \Omega \to \Omega$, one gets a complete Heyting subalgebra $\Omega_j$ of $\Omega$, i.e. the object of $j$-closed truth values. On the other hand, given $\iota: \Omega' \hookrightarrow \Omega$ complete Heyting subalgebra it seems this has a right adjoint $\kappa$ which post-composed with the inclusion yields a map $j:=\iota\kappa:\Omega \to \Omega$ with the following properties:

1. $j(\top) =\top$ since right adjoints preserve limits and $\iota$ is a morphism of complete Heyting algebras,
2. $jj = j$ because $j$ is extensive (by (1)) and a comonad,
3. $j$ is order preserving since both $\iota$ and $\kappa$ are (or, alternatively: $j$ is left exact since both $\iota$ and $\kappa$ are).

Matteo Capucci (he/him) (Jan 19 2021 at 10:31):

I've never seen this spelled out explicitly though so I might be tricking myself

Fawzi Hreiki (Jan 19 2021 at 12:13):

Invoking completeness somehow feels wrong because it's not a first-order property. A better approach (imo) is just to talk about co-reflective subalgebras, in which case its obviously true since every co-localisation induces an idempotent co-monad and vice versa.

Fawzi Hreiki (Jan 19 2021 at 12:17):

It may be interesting to ask about internal completeness (i.e. when the subalgebra, regarded as an internal category, has all $\mathscr{E}$-indexed limits).

Fawzi Hreiki (Jan 19 2021 at 12:22):

Just doing a quick google search, it seems theres this paper which proves an adjoint functor theorem for internal categories which seems relevant to this.

Matteo Capucci (he/him) (Jan 19 2021 at 12:52):

You're right about completeness.
I might have to go galaxy brain and really work with internal categories, but it seems unlikely in my case... I have to meditate

Fawzi Hreiki (Jan 19 2021 at 12:58):

Apart from some axiom of choice stuff, categories internal to toposes are not so different from small categories. In this case in particular, using internal categories seems totally natural since there is a correspondence between sites in $\mathscr{E}$ and internal categories in $\mathscr{E}$ equipped with modalities on their presheaf category.

Fawzi Hreiki (Jan 19 2021 at 12:59):

Which is just the usual relationship between Grothendieck topologies and modalities but relativised to $\mathscr{E}$.