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Morgan Rogers (he/him) (Sep 08 2026 at 12:37):I've just come up with a construction that takes two frames and constructs a new frame by identifying the top of with the bottom of (everything in is above everything in ). The "obvious" joins and meets seem to distribute just fine.
Has anyone encountered this before? Perhaps on spaces? It's a construction that seems reasonable from the algebraic/frame point of view but counterintuitive from the topological point of view.
Chris Grossack (she/they) (Sep 08 2026 at 15:56):This is a well known operation in the theory of posets/lattices, called the "ordinal sum" or the "ordered sum" or the "linear sum". See, eg, Section 1.24 of my edition of Davey and Priestley's "Introduction to Lattices and Order".
It's definitely "well known" that the linear sum of two distributive lattices is distributive, and people definitely study this (though I'm not sure of any references off the top of my head). I'm not sure of anybody using this notion for frames (and I can't find anything on quickly looking around for it either), but lattice-theoretically it's a very natural thing to consider.
Chris Grossack (she/they) (Sep 08 2026 at 16:00):Do you know what it corresponds to for (topological) spaces, even? Say we have two spaces X and Y and we consider the new space (hopefully it has enough points?) we get from doing this operation to their opens (say Y on top of X)... It should hopefully be pretty easy to figure out what the resulting space is, in terms of X and Y? I'm busy just this second, but I might think about it later today if someone else doesn't think about it first
fosco (Sep 08 2026 at 17:15):I don't know if it has a name, but a back-of-the-envelope computations seems to say that if you perform this construction on two spaces, (the result is spatial and) the topology is on the disjoint union of their carriers, one inclusion (the space above) is a closed map, the other (the space below) is open.
James Deikun (Sep 08 2026 at 17:48):Is this true? If the topology is spatial, then every open set that contains a point of the space above must contain all points of the space below.
Morgan Rogers (he/him) (Sep 08 2026 at 18:17):It is true that there are open and closed subspaces as fosco describes whose points correspond to the original spaces, but the standard way to achieve that is Artin gluing... so I guess this is a quotient of the Artin gluing?
Morgan Rogers (he/him) (Sep 09 2026 at 07:25):(deleted)
Graham Manuell (Sep 09 2026 at 10:06):Morgan Rogers (he/him) This is exactly an Artin gluing, with the meet semilattice map given by h(a) = [a=1], I think, i.e. h(a) = 1 if a = 1 and h(a) = 0 otherwise.
Graham Manuell (Sep 09 2026 at 10:14):By the way, the operation on posets is also studied from a categorical perspective in @Roy Ferguson's PhD thesis.
Morgan Rogers (he/him) (Sep 09 2026 at 10:24):Oh of course, you have to choose which meet semilattice map to glue along. Thanks @Graham Manuell !
Jonas Frey (Sep 09 2026 at 11:34):I would have said that artin gluing in the sense of toposes puts the toposes "on top of each other" in a sense, but on the level of points rather than frames of opens ... In particular gluing to localic toposes along the constant 1 functor should give something like the join/ordinal sum of points (with specialization ordering).
Morgan Rogers (he/him) (Sep 09 2026 at 12:08):Where can I find that thesis, @Roy Ferguson ?
Graham Manuell (Sep 09 2026 at 17:16):https://scholar.sun.ac.za/server/api/core/bitstreams/29b72861-2a64-469b-94f9-8c2c817f02a8/content
Roy Ferguson (Sep 10 2026 at 11:30):Sorry I am late to the conversation and thank you @Graham Manuell for sharing! We call this operation "stacking of spaces" and arrived at a special case of it by transporting stacking of preorders along the specialisation equivalence. But this operation works out for Top as well. All these operations are examples of monoidal sum structures (in the sense of Zurab Janelidze), which have the property that a certain nice class of morphisms called the c-reflecting morphisms are all extensive (in the sense Michael Hoefnagel and Emma Theart). Curiously, the usual coproduct and stacking of spaces are the only monoidal sum structures on Top where this is the case (see Section 4.5).
Sridhar Ramesh (Sep 10 2026 at 13:40):I just want to point out that, pedantically, what is described in the OP is perhaps not quite the ordinal sum as mentioned in post 2, because it sounds as though the OP identifies one frame's top element with another's bottom element, instead of keeping those two distinct.
Morgan Rogers (he/him) (Sep 10 2026 at 15:01):That's a helpful caveat @Sridhar Ramesh. Without the identification I described, the locale has an extra point "in the middle" which does not come from either of the original spaces (but I'm fairly sure that the operation discussed in the remaining posts is the one I described)
Vincent Moreau (Sep 10 2026 at 17:37):This discussion is fascinating! And just to add to the caveat, although the ordinal sum of posets has a monoidal unit, its restriction to frames does not as the unit would have to be empty, while the stacking of spaces is a legit monoidal struture with unit
Roy Ferguson (Sep 11 2026 at 06:42):@Vincent Moreau yes! This was a frustration of ours when looking for sum structures in categories of relational structures bounded above and below. Monoidal sum structures always have unit the initial object, and so stacking/ordinal sum is not a candidate for such categories. But, if we "transport" stacking along the functor Top->Frm, I think this operation actually gives us a monoidal product structure, where the unit is terminal.
Another thing I didn't mention regarding Artin gluings is that stacking and coproduct in Top (the sufficiently extensive ones) are two extremes of Artin gluings, where for coproduct you glue along the map that sends everything to the top, while for stacking you glue along the map that only sends top to top and everything else to bottom.