Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: deprecated: topos theory

Topic: Basic questions

Eduardo Ochs (Sep 07 2020 at 02:22):

Is it really ok to ask basic questions here? Let me try...

Suppose that we have a topos $\mathcal{E}$ with a (Lawvere-Tierney) topology $j:\Omega \to \Omega$. Then for any subobject $s: S \rightarrowtail A$ in it we can construct its closure by building its characteristic arrow $\chi_s$, composing it with $j$, and pulling back the arrow $t:1 \to \Omega$ along $j \circ \chi_s$, as in the diagram below.

Eduardo Ochs (Sep 07 2020 at 02:22):

closure.png

Eduardo Ochs (Sep 07 2020 at 02:23):

``Everybody knows'' that we will then have $s \subseteq \overline{s}$, but I realized that I never knew the details of this, and I'm trying to fill the gaps. I'm trying to find an explicit construction for the monic $S \rightarrowtail \overline{S}$ and I'm failing miserably... Anyone knows how to do that? This is theorem 21.1 of McLarty's "Elementary Categories, Elementary Toposes", but his proof is too terse for me...

David Michael Roberts (Sep 07 2020 at 07:52):

@Eduardo Ochs you do have $j\circ true= true\colon 1\to \Omega$, so there should be a commuting square with $S \to 1$ along the top, $S\to A$ on the left, $A\to \Omega \to \Omega$ along the bottom, and $true\colon 1\to \Omega$ on the right. Then I think the universal property of $\overline{S}$ as a pullback gives $S\to \overline{S}$, no?

David Michael Roberts (Sep 07 2020 at 07:54):

More simply, take your diagram, add the identity arrow from 1 to 1, and then apply the pullback's universal property!

Eduardo Ochs (Sep 07 2020 at 16:26):

Ahaaa! Beatiful! Thanks!!!

Eduardo Ochs (Sep 08 2020 at 02:54):

Here is the finished version:

Eduardo Ochs (Sep 08 2020 at 02:54):

mclarty_theorem_21.1_p1.png

Eduardo Ochs (Sep 08 2020 at 02:54):

mclarty_theorem_21.1_p2.png

Morgan Rogers (he/him) (Sep 08 2020 at 10:33):

That's a nice neat write-up :heart_eyes:

Eduardo Ochs (Sep 21 2020 at 03:54):

Hi, another question... I am working in toposes of the form $\mathbf{Set}^\mathbf{C}$, where $\mathbf{C}$ is a finite poset, and I am in a situation in which for any two subobjects $P$ and $Q$ of 1 in the topos obeying $P \le Q$ the closure of the monic $p: P \rightarrowtail Q$ is easy to calculate; to be more precise, closures of these subobjects can be calculated using the operations in the paper Planar Heyting Algebras for Children 2 (PDF here), and so they are easy to visualize.

Eduardo Ochs (Sep 21 2020 at 03:54):

I want to prove that for any monic $d: D \rightarrowtail E$ in that topos its closure, $\overline{d}: \overline{D} \rightarrowtail E$, is equal to the union of things that are easy to calculate and to visualize, in the following sense...

Eduardo Ochs (Sep 21 2020 at 03:55):

I know how to express the object $E$ as a union of representables. Formally, this "...as a union of representables" is a pair made of an index set $K$ and a family of monics, $(r_k: R_k \rightarrowtail E)_{k \in K}$, such that each $R_k$ is a (representable) subobject of 1, and $\bigcup_{k \in K} \text{image}(r_k) = E$. And for each $k \in K$ I can form a diagram like this:

Eduardo Ochs (Sep 21 2020 at 03:56):

r_k.png

Eduardo Ochs (Sep 21 2020 at 03:57):

Is it possible to prove that $\bigcup_{k \in K} \text{image} (r_k \circ \overline{r_k^{-1}(d)}) = \text{image}(\overline{d})$? I am still struggling with the details on Lawvere-Tierney Topologies in the first pages about them in each of a handful of books... I haven't reached yet the parts in which the `$j$'s interact with unions of subobjects -- so brief hints like "look at page xxx in book yyy" are probably going to be very useful.

Eduardo Ochs (Sep 21 2020 at 03:57):

Thanks in advance!!!

Eduardo Ochs (Sep 21 2020 at 05:17):

Update: I think I found an argument that works! Treating all entities from here on as elements of $\text{Sub}(E)$, we have: $r_k^{-1}(D) = R_k \cap D$, $\overline{r_k^{-1}(D)} = R_k \cap \overline{D}$, $\bigcup_k R_k = E$; so $\bigcup_k (R_k \cap \overline{D}) = (\bigcup_k R_k) \cap \overline{D} = E \cap \overline{D} = \overline{D}$...

Morgan Rogers (he/him) (Sep 21 2020 at 11:15):

1. The closure operator is "universal", so stable under pullback, in the sense that $\overline{r_k^{-1}(d)} = r_k^{-1}(\overline{d})$, as you note in your diagram.
2. The closure operator is a complete Heyting algebra homomorphism, which is to say that $\overline{\bigcup_{k \in K} r_k} = \bigcup_{k \in K} \overline{r_k}$. Note that I'm in the habit of conflating a monomorphism with the subobject it represents, so I've omitted "image" here.

It seems to me that the argument in your Update uses these facts appropriately, so it works!

Eduardo Ochs (Nov 24 2020 at 07:46):

Hi all, especially @David Michael Roberts and @[Mod] Morgan Rogers,

I finished writing down my notes on topologies and closure operators - they are here: http://angg.twu.net/math-b.html#clops-and-tops - but I added a section that is still unfinished on something that I guess that is known, but I can't find it in the literature... namely, that for toposes of the form $\mathbf{Set}^\mathbf{D}$, where this $\mathbf{D}$ is a finite poset, the operation that takes a closure operator on the topos and restricts it to $\mathrm{Sub}(1)$ is a bijection between topologies and operations on the Heyting Algebra of the truth-values of the topos that obey $P \le P^* = P^{**}$ and $(P \land Q)^* = P^* \land Q^*$...

Any pointers would be very welcome! =)

Morgan Rogers (he/him) (Nov 24 2020 at 09:10):

Sounds like the corresponding Lawvere-Tierney topology to me.

Eduardo Ochs (Nov 24 2020 at 09:39):

Corresponding to what?...

Eduardo Ochs (Nov 24 2020 at 09:43):

Every closure operator has a corresponding Lawvere-Tierney topology and vice-versa, but these operators on the Heyting Algebra of truth-values are a third thing. I am not sure if I understood you correctly, though.

Jens Hemelaer (Nov 24 2020 at 09:49):

The corresponding closure operator on $\mathrm{Sub}(1)$ is called a nucleus. For localic Grothendieck toposes, Lawvere–Tierney topologies (equivalently, Grothendieck topologies on $\mathbf{D}^\mathrm{op}$) correspond bijectively to nuclei on the frame $\mathrm{Sub}(1)$. This is also the case here, because $\mathbf{Set}^\mathbf{D}$ is a localic topos for any poset.

Jens Hemelaer (Nov 24 2020 at 09:57):

In the case where $\mathbf{D}$ is a finite poset, these correspond precisely to the subsets of $\mathbf{D}$.

In "Grothendieck topologies on a poset" by @Bert Lindenhovius it is shown that more generally the Grothendieck topologies on any Artinian poset $\mathbf{P}$ correspond to the subsets of $\mathbf{P}$.

Jens Hemelaer (Nov 24 2020 at 10:03):

As a reference:
In Maclane–Moerdijk, "Sheaves in Geometry and Logic", IX.5, Corollary 6, it is shown that nuclei on a locale $X$ correspond to Lawvere–Tierney topologies on the topos $\mathbf{Sh}(X)$.

Eduardo Ochs (Nov 24 2020 at 10:13):

Wow!!!! Thanks!!!!!! =) =) =)

Morgan Rogers (he/him) (Nov 24 2020 at 10:39):

Sorry, just saying "Lawvere-Tierney topology" admittedly wasn't so helpful. What I was trying to say was that the LT topology is a mapping $j: \Omega \to \Omega$, and in a presheaf topos the global elements of $\Omega$, $\mathrm{Hom}(1, \Omega) = \Gamma(\Omega)$ are precisely the subterminal objects, which is to say the truth values of the topos, so the nucleus is given by $(j \circ -): \Gamma(\Omega) \to \Gamma(\Omega)$.

Matteo Capucci (he/him) (Nov 24 2020 at 10:43):

I understand the confusion, I routinely call nuclei LT-topologies and viceversa :laughing:

Fawzi Hreiki (Nov 24 2020 at 13:28):

It should generally be harmless to confuse the two right?

Fawzi Hreiki (Nov 24 2020 at 13:28):

Since they’re both modal operators on the truth values of the category - just one is internal and one is external

Matteo Capucci (he/him) (Nov 24 2020 at 13:28):

Yeah, I guess

Fawzi Hreiki (Nov 24 2020 at 13:29):

And when the category is a topos, an external operator will internalise to an internal one (like all the other logic which internalises)

Matteo Capucci (he/him) (Nov 24 2020 at 13:35):

I wouldn't use the term Lawvere-Tierney topology if the category wasn't a topos. Maybe one should say 'internal nucleus'

Fawzi Hreiki (Nov 24 2020 at 13:39):

I think Johnstone calls Lawvere-Tierney topologies ‘local operators’

Fawzi Hreiki (Nov 24 2020 at 13:41):

I don’t really see the problem with using the same name for both since that’s what happens with unions, intersections, quantifiers, etc

Matteo Capucci (he/him) (Nov 24 2020 at 13:45):

Ok but 'topology' doesn't really make sense outside a topos. Of course the two notions are the same, syntactically speaking, but LT topology has a different attitude.

Jens Hemelaer (Nov 24 2020 at 14:15):

Fawzi Hreiki said:

It should generally be harmless to confuse the two right?

For localic toposes, it's more or less the same thing. For general Grothendieck toposes, each Lawvere–Tierney topology defines a nucleus on $\Gamma(\Omega)$ as @[Mod] Morgan Rogers says, but different Lawvere–Tierney topologies can induce the same nucleus on $\Gamma(\Omega)$. This is maybe why the terminology 'nucleus' is used for frames and the terminology 'Lawvere–Tierney topology' only for toposes. I agree that they are very related concepts.

Morgan Rogers (he/him) (Nov 24 2020 at 14:16):

Indeed, in an extreme case, like the topos of presheaves on a monoid, all non-degenerate topologies induce the same nucleus :upside_down:

David Michael Roberts (Nov 24 2020 at 21:56):

It's good you are writing this in detail! I recently finished supervising a student who was proving some category theoretic stuff in full detail, and it honestly surprised me how much I would have left out if I were writing it myself. Unfortunately I don't know of anything in the literature about this, sorry about that.