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Stream: learning: questions

Topic: modular lattices

sarahzrf (Nov 26 2020 at 19:18):

is there some sort of enriched notion generalizing that of a modular lattice? i've seen the law μ(A) + μ(B) = μ(A ∪ B) + μ(A ∩ B) referred to as "modularity" by at least one source, and it seems an awful lot like this is in reference to modular lattices

sarahzrf (Nov 26 2020 at 19:19):

indeed, i half figured out how it would need to work and then ran into the other half near the end of Taking Categories Seriously: image.png image.png

sarahzrf (Nov 26 2020 at 19:22):

image.png

sarahzrf (Nov 26 2020 at 19:25):

we have

$$\mu(x \lor y) = \mu(\bot, x \lor y) = \mu(\bot, x \land y) + \mu(x \land y, x) + \mu(x, x \lor y),$$

so

$$\mu(x \lor y) + \mu(x \land y) = \mu(\bot, x \land y) + \mu(\bot, x \land y) + \mu(x \land y, x) + \mu(x, x \lor y)$$

sarahzrf (Nov 26 2020 at 19:28):

if $\mu(x, x \lor y) = \mu(x \land y, y)$, which certainly seems like a kind of "modularity" condition since those are sort of "two sides of the square", then this gives us

$$\mu(x \lor y) + \mu(x \land y) = \mu(\bot, x \land y) + \mu(x \land y, x) + \mu(\bot, x \land y) + \mu(x \land y, y) = \mu(x) + \mu(y)$$

sarahzrf (Nov 26 2020 at 19:29):

so: is there some generalized notion of "modularity" which gives rise to both modular lattices and this condition?

Nathanael Arkor (Nov 26 2020 at 19:32):

Just an observation that modularity looks like the condition imposed in a linearly distributive category. In particular, in a bicartesian category, we have a canonical linearly distributive category given by coproduct and product, and we can ask for this to be an isomorphism, which presumably gives a categorification of modular lattices (though the nLab notes that when the category is distributive, this means such a "modular category" must be a preorder).

sarahzrf (Nov 26 2020 at 19:34):

hmm, what similarity do you see?

sarahzrf (Nov 26 2020 at 19:35):

i'm not really super familiar with either concept tbh—i only looked up modular lattices again the other day because i thought i remembered they might be relevant to the thoughts leading to this question

Nathanael Arkor (Nov 26 2020 at 19:36):

I was essentially just musing that if modular lattices could be described as certain polycategories, whose modular law arose from polycategorical composition, then maybe if this other modularity condition is related to modular lattices, it too could be described as an appropriate composition condition.

Nathanael Arkor (Nov 26 2020 at 19:37):

But it's a very long way from a concrete or helpful answer :grinning_face_with_smiling_eyes:

sarahzrf (Nov 26 2020 at 19:39):

oh no i meant very literally like

sarahzrf (Nov 26 2020 at 19:39):

im looking at the data of a linearly distributive category, and i don't rly quite see a resemblence to modular lattices

sarahzrf (Nov 26 2020 at 19:39):

like, which formulations of each do you think look similar?

Nathanael Arkor (Nov 26 2020 at 19:47):

The modular law says that a ≤ b implies a + (c * b) = (a + c) * b for all c. A (strong) linearly distributive category has an isomorphism $$a \otimes (c \parr b) \cong (a \otimes c) \parr b$$. So the form of the equation is essentially the same, it's just missing the condition about a morphism $a \to b$ (which is natural condition to impose particularly in the bicartesian setting, when such a map induces a linear distribution).

Fabrizio Genovese (Nov 26 2020 at 22:46):

A lattice is modular iff it omits $N_5$. Maybe we could try to find a categorical equivalent of $N_5$ and "omits"?

sarahzrf (Nov 26 2020 at 22:51):

that sounds dubious to me

sarahzrf (Nov 26 2020 at 22:52):

omitting N₅ strikes me as a "coincidence" kind of characterization of modularity, the kind of thing that's sensitive to changing setting

Fabrizio Genovese (Nov 26 2020 at 22:53):

I'm not sure, there's a lot of classical universal algebra developed in that way

Fabrizio Genovese (Nov 26 2020 at 22:53):

E.g. "omits $N_5$ + omits $M_3$" gives you distributivity

Fabrizio Genovese (Nov 26 2020 at 22:54):

You can also go meta and look at omissions in the lattice of congruences of an algebra, and ask things like "What are all the algebraic varieties such that algebras of those varieties have lattices of congruences always omitting some lattice?"

Fabrizio Genovese (Nov 26 2020 at 22:55):

This is basically Mal'Cev theory and there are tons of open problems there.

Fabrizio Genovese (Nov 26 2020 at 22:57):

(This is, btw, the part of Lawvere theories that really doesn't satisfy me much. There's a lot of stuff that can be stated as I did above, more or less, of which I don't know if a decent categorification exists. What is the lattice of congruences of an algebra from the point of view of Lawvere theories?)

Nathanael Arkor (Nov 26 2020 at 23:02):

You may be interested in Hoefnagel–Jacqmin–Janelidze's The matrix taxonomy of left exact categories, which examines classes of categories defined by Mal'tsev-like conditions.

Fabrizio Genovese (Nov 26 2020 at 23:09):

This seems interesting. So they generalize classes of varieties linked by Mal'cev conditions to categories of left-exact categories satisfying some internal relations

Fabrizio Genovese (Nov 26 2020 at 23:09):

Cool

Fabrizio Genovese (Nov 26 2020 at 23:11):

Oh no wait, they have classes of left-exact categories, not categories of left exact categories