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Stream: theory: category theory

Topic: subtraction

sarahzrf (Jul 29 2021 at 15:41):

Why does subtraction categorify so poorly?

sarahzrf (Jul 29 2021 at 15:42):

ive read a couple of things now where we want to view a ring or group as a decategorification of something or other, and it seems the generally-accepted solution—at least insofar as forming a grothendieck group/ring is a generally-accepted solution—is simply to add formal inverses

sarahzrf (Jul 29 2021 at 15:43):

this strikes me as unsatisfying—i feel like if what you want is to categorify a ring or group, then the subtraction or inversion should themselves categorify?

sarahzrf (Jul 29 2021 at 15:43):

idk

sarahzrf (Jul 29 2021 at 15:43):

what's going on here?

Mike Shulman (Jul 29 2021 at 16:36):

Forming fibers or cofibers in a pointed category (and particularly a stable $\infty$-category) is often a reasonable categorification of "subtraction".

sarahzrf (Jul 29 2021 at 19:04):

Mike Shulman said:

Forming fibers or cofibers in a pointed category (and particularly a stable $\infty$-category) is often a reasonable categorification of "subtraction".

how so?

Mike Shulman (Jul 29 2021 at 19:26):

Well, for example, the Euler characteristic maps cofibers to subtraction: $\chi(X/A) = \chi(X) - \chi(A)$.

Ian Coley (Jul 29 2021 at 20:19):

For another thing: consider a pointed category with pushouts. That lets us form coproducts:

image.png

If we then compute the cofibre $C$ of $Y\to X\vee Y$, that's another pushout glued onto the first one:

image.png

But since these are two pushouts stacked on top of each other, the outside square is also a pushout. We conclude that $C\cong X$, so the cofibre of the natural inclusion map $Y\to X\vee Y$ is just $X$.

sarahzrf (Jul 29 2021 at 20:53):

huhh

sarahzrf (Jul 29 2021 at 20:53):

how are pushouts computed in $\mathrm{Set}_*$, say

Mike Shulman (Jul 29 2021 at 20:54):

They're just pushouts in Set, with the induced basepoint.

sarahzrf (Jul 29 2021 at 20:59):

ooh, does the forgetful functor create connected colimits or sth

Mike Shulman (Jul 29 2021 at 21:15):

sth

sarahzrf (Jul 29 2021 at 21:28):

aww

Jon Awbrey (Jul 29 2021 at 22:36):

Cf: Animated Logical Graphs • 75

A couple months ago I was trying to come up with a good name for an operation dual to logical implication in a particular context, and what I got happened to be subtraction. See the above blog post. So maybe propositions as types would suggest a cat-analogue?

Henry Story (Jul 30 2021 at 04:42):

Yes, I have heard of subtractions as co-implications in "complement topoi"
They come up in more recent work on bi-intutionism such as this recent paper "Pragmatic and dialogic interpretations of bi-intutionism". see this recent tweet in a thread on dualities started by John Baez. I am not sure if this is the same idea of subtraction that @sarahzrf is looking for.

@johngstell @johncarlosbaez @giuliasindoni Ah yes, thanks. A later paper: "Pragmatic and dialogic interpretations of bi-intuitionism" In short: When both sides of the duality are combined so as not to collapse into first order logic, a logic of dialogue appears https://apcz.umk.pl/czasopisma/index.php/LLP/article/view/LLP.2014.011 https://twitter.com/bblfish/status/1413466013488799745/photo/1

- The 🐟‍‍ BabelFish (@bblfish)

Henry Story (Jul 30 2021 at 04:48):

In particular from Pragmatic and dialogic interpretations of bi-intutionsm. Part I

It is very appropriate to ask such a question about bi-intuitionism: following Rauszer’s approach researchers in this area usually define bi- intuitionism by extending intuitionistic logic with the connective of subtraction C \ D, to be read as “C excludes D”, which in algebraic terms is the left adjoint to disjunction in the same way as implication is the right adjoint to conjunction (see the rules in (1.1) below). This pair of adjunctions establishes a duality between the core minimal fragments of intuitionism and co-intuitionism, namely, intuitionistic conjunction and implication with a logical constant for validity, on one hand, and co- intuitionistic disjunction and subtraction with invalidity, on the other.

(I could not find Part II last time I looked)

Mike Shulman (Jul 30 2021 at 04:59):

Another context in which "disjunction" is a two-variable right adjoint is a $\ast$-autonomous category.

Morgan Rogers (he/him) (Jul 30 2021 at 10:09):

Ian Coley said:

If we then compute the cofibre $C$ of $Y\to X\vee Y$...

Why is this given a different name in a pointed category versus any category with a terminal object (where it's called a cokernel)? It doesn't seem quite dual to the notion of fiber, at least morally, since the whole point of a fiber is that it extracts information over a particular global point $1 \to X$, and that becomes a lot less meaningful in a pointed category where there is exactly one such morphism.

Matteo Capucci (he/him) (Jul 30 2021 at 10:27):

my favourite categorification of subtraction (or, more generally, inverse operation of a monoid) is through compact closed categories.
explicitly, if a monoid categorifies to a monoidal category, a group should categorify to a compact closed category.
in fact the property '$a + b = c$ iff $a = c - b$', which defines substraction, gets categorified to the adjunction $[a \otimes b, c] \cong [a, c \otimes b^*]$. Or more simply, the unit and counit associated to the dual of an object categorify the properties $b-b=0$ and $-b + b = 0$.

Matteo Capucci (he/him) (Jul 30 2021 at 10:29):

also the int construction fits into this picture, as a categorification of the group-completion (idk if this is the correct name) of a monoid

Henry Story (Jul 30 2021 at 12:02):

If the left adjoint on + gives us subtraction, is there an adjoint that gives us division?

Simon Burton (Jul 30 2021 at 12:11):

This is reminding me of the relation between a group and a torsor for the group: the torsor is where you get to do division (or subtraction) and that gets you back to the group. Example: in an affine space you can subtract one point from another, and this gives you an element of the translation group.

Now I'm wondering about the connection to the basepoint discussion above: a group is a torsor with a basepoint (or, a torsor is a group that forgot its identity element). :thinking:

Jacques Carette (Jul 30 2021 at 12:35):

One 'problem' with the int construction is that it doesn't work in the context of rig categories: if you perform int on $+$, you collapse everything. See Ring completion of rig categories for how involved this gets. If nothing else, everyone should read Thomason's wonderful (and short - 4 pages)Beware the Phony Multiplication on Quillen's $\mathscr{A}^{-1}\mathscr{A}$ .

Cole Comfort (Jul 30 2021 at 13:20):

Simon Burton said:

This is reminding me of the relation between a group and a torsor for the group: the torsor is where you get to do division (or subtraction) and that gets you back to the group. Example: in an affine space you can subtract one point from another, and this gives you an element of the translation group.

Now I'm wondering about the connection to the basepoint discussion above: a group is a torsor with a basepoint (or, a torsor is a group that forgot its identity element). :thinking:

Because you do not need constants to define a torsor in terms of a ternary operation, if you admit the empty torsor, then torsors are "groups" with possibly no identity at all!

Matteo Capucci (he/him) (Jul 30 2021 at 14:27):

Henry Story said:

If the left adjoint on + gives us subtraction, is there an adjoint that gives us division?

you need to use $\times$ instead

Fawzi Hreiki (Jul 30 2021 at 14:43):

Its the right adjoint to exponentiation

Fawzi Hreiki (Jul 30 2021 at 14:44):

'Aspects of fractional exponent functors'

Jon Awbrey (Jul 30 2021 at 15:24):

Fawzi Hreiki said:

Its the right adjoint to exponentiation

Fawzi Hreiki said:

'Aspects of fractional exponent functors'

But exponentiation is implication, sorta?

Ian Coley (Jul 30 2021 at 15:40):

Morgan Rogers (he/him) said:

Ian Coley said:

If we then compute the cofibre $C$ of $Y\to X\vee Y$...

Why is this given a different name in a pointed category versus any category with a terminal object (where it's called a cokernel)? It doesn't seem quite dual to the notion of fiber, at least morally, since the whole point of a fiber is that it extracts information over a particular global point $1 \to X$, and that becomes a lot less meaningful in a pointed category where there is exactly one such morphism.

I guess it really is a cokernel; I was just trying to draw out the un-stable version of Mike's point above mine.

Jon Awbrey (Jul 30 2021 at 15:48):

Then again, in differential logic, which is basically differential geometry over GF(2), algebraic $+$ corresponds to logical $\mathrm{xor}$ and set-theoretic symmetric difference, so $+ = -$ .

Mike Shulman (Jul 30 2021 at 17:02):

@Morgan Rogers (he/him) The word cofiber isn't specific to a pointed category; it's just a different name for cokernel. The word "cofiber" tends to be used in more topological and homotopy-theoretic contexts, while "cokernel" tends to be used in more additive contexts. In contexts that are both homotopy-theoretic and additive, like spectra or chain complexes, it's a toss-up, and in chain complexes you also have a third word "cone" for the same thing.

Mike Shulman (Jul 30 2021 at 17:02):

The same is true of "fiber" and "kernel".

Mike Shulman (Jul 30 2021 at 17:03):

It's true that fibers also exist in unpointed categories, where you have to choose a basepoint to take the fiber over. But they are very interesting and useful in pointed categories, where they are the precise dual of a cofiber/cokernel.

Mike Shulman (Jul 30 2021 at 17:04):

By duality, one might argue that in an unpointed category the appropriate notion of "cofiber/cokernel" would be a pushout along a (chosen!) map to the initial object.

Jon Awbrey (Jul 30 2021 at 17:10):

Evidently we have a smörgåsbordism on offer ...

Morgan Rogers (he/him) (Jul 31 2021 at 09:04):

Mike Shulman said:

By duality, one might argue that in an unpointed category the appropriate notion of "cofiber/cokernel" would be a pushout along a (chosen!) map to the initial object.

I would indeed argue that it would be useful to have a terminological distinction here. Namely:
kernel: pullback along unique map from initial object
fiber: pullback along specified map from terminal object
and similarly for the dual concepts,
cokernel: pushout along unique map to the terminal object
cofiber: pushout along specified map to the initial object

Morgan Rogers (he/him) (Jul 31 2021 at 10:15):

That said, I can't think of (m)any common non-pointed categories where this notion of cofiber would be interesting, so the conflation that currently happens is harmless.

Fawzi Hreiki (Jul 31 2021 at 11:28):

The category of rings is an interesting example of a non-pointed category whose initial object is not strict.

Fawzi Hreiki (Jul 31 2021 at 11:30):

Cofibers of course then correspond to fibers over $\text{Spec}\mathbb{Z}$-points in $\text{AffSch}$.

Fawzi Hreiki (Jul 31 2021 at 11:32):

Usually, in the non-Abelian case, especially in geometry, fiber has the more general meaning of pullback over any map at all.