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Stream: theory: mathematics

Topic: A couple of conjectures on heaps

John Baez (Jun 28 2024 at 12:37):

I hadn't meant to get drawn in this deep into the study of [[heaps]], but some people are posing nice conjectures about them after I posted about them on Mastodon, so I feel a duty to record those conjectures.

John Baez (Jun 28 2024 at 12:42):

First, while I believe the category of heaps is equivalent to the category of group-torsor pairs, there are some questions about the nature of the latter category: what it actually is, and whether it can be defined using the Grothendieck construction.

I think the category of group-torsor pairs, say $\mathsf{Tors}$, works like this:

• an object $(G,H)$ is a group $G$ together with a $G$-torsor $H$, i.e. a nonempty set $H$ equipped with a free and transitive $G$-action.
• a morphism $(\phi, f) : (G,H) \to (G',H')$ is a group homomorphism $\phi: G \to G'$ and a set map $f: H \to H'$ such that $f(gh) = \phi(g) f(h)$ for all $g \in G$, $h \in H$.

John Baez (Jun 28 2024 at 12:45):

It would be nice to verify that this really is equivalent to the category of heaps as claimed in the nLab article [[heap]]. There shouldn't be anything very hard about checking this.

But then there's the question of how this category $\mathsf{Tors}$ is related to all the categories $G \mathsf{Tors}$ of $G$-torsors. In $G \mathsf{Tors}$

• an object is a $G$-torsor $H$ and
• a morphism $f: H \to H'$ is a set map such that $f(g h) = g f(h)$.

John Baez (Jun 28 2024 at 12:49):

LeeMph raised this question:

Question. Is there a pseudofunctor from $\mathsf{Grp}$ to $\mathsf{Cat}$ sending each group $G$ to the category $G \mathsf{Tors}$, either covariant or contravariant, such that performing the Grothendieck construction stitches together all the category $G \mathsf{Tors}$ to form the category $\mathsf{Tors}$?

John Baez (Jun 28 2024 at 12:51):

We discussed it a while, and his latest verdict is negative, but he thinks something a bit fancier will work. So a better version of the question above may be:

Question. How exactly can we get $\mathsf{Tors}$ from all the categories $G \mathsf{Tors}$?

John Baez (Jun 28 2024 at 12:55):

By the way, if anyone had the insane amount of energy necessary to do this, it would be nice to investigate this in a general topos, since the concept of 'torsor' becomes very interesting then: it more or less subsumes the concept of [[principal bundle]], which is very important in geometry.

John Baez (Jun 28 2024 at 12:56):

Indeed maybe someone has already studied it at that level of generality!

Rémy Tuyéras (Jun 28 2024 at 12:58):

Here are some potential directions (which I did not have time to check thoroughly yet, but just wanted to share some intuitions as I have to go):

• can we define a $G$-torsor as a functor $T_G \to \mathbf{Set}$ where $T_G$ is the limit sketch (Lawvere theory?) made of Cartesian products $G^{n}$ and group homomorphism between them?
• (if yes then) denoting $\mathsf{Mod}(T_G) \hookrightarrow [T_G,\mathbf{Set}]$ the reflective subcategory restricted to limit preserving functors, is the mapping $\mathcal{T}:G \mapsto \mathsf{Mod}(T_G)$ a functor $\mathsf{Grp} \to \mathbf{Cat}$ (or $\mathsf{Grp} ^{\mathsf{op}}\to \mathbf{Cat}$)?
• (if yes then) is $\mathsf{Tors}$ the Grothendieck construction of $\mathcal{T}$?

John Baez (Jun 28 2024 at 13:05):

Second - sorry, I'm just plowing ahead here before I forget - there's the question of what's the free group on a heap. There's a forgetful functor

$U : \mathsf{Grp} \to \mathsf{Heap}$

sending any group $G$ to the underlying set of $G$ made into a heap with operation $t(a,b,c) = a b^{-1} c$. This must have a left adjoint (by general nonsense about Lawvere theories), so:

Question. Concretely as possible, what is the left adjoint of $U : \mathsf{Grp} \to \mathsf{Heap}$?

And Oscar Cunningham proposed an answer. First, recall that there's a functor

$\mathrm{Str}: \mathsf{Heap} \to \mathsf{Grp}$

sending any heap $H$ to the group of all maps $h \mapsto t(a,b,h)$ from $H$ to itself, where $a,b$ are arbitrary elements of $H$. (If we think of a heap as a group-torsor pair, this functor picks out the group in that pair.)

I foolishly ventured that the left adjoint of $U : \mathsf{Grp} \to \mathsf{Heap}$ was this functor $\mathsf{Str}$, but Oscar improved that guess:

Conjecture. The left adjoint of $U : \mathsf{Grp} \to \mathsf{Heap}$ sends any heap $H$ to the group $\mathbb{Z} \ast \mathsf{Str}(H)$, where $\ast$ is the "free product", or coproduct, of groups.

John Baez (Jun 28 2024 at 13:09):

He motivated this with a nice analogy: the left adjoint of the forgetful functor from real vector spaces to affine spaces takes any affine space $\mathbb{R}^n$ to $\mathbb{R}^{n+1}$, where the extra dimension arises from freely putting in an origin.

Rémy Tuyéras (Jun 28 2024 at 13:38):

Rémy Tuyéras said:

• can we define a $G$-torsor as a functor $T_G \to \mathbf{Set}$ where $T_G$ is the limit sketch (Lawvere theory?) made of Cartesian products $G^{n}$ and group homomorphism between them?

I will just adjust this statement by proposing to take $T_G$ to be the Lawvere theory (EDIT: product sketch):

• whose objects are the Cartesian products $G^0$ and $G^1$
• and whose morphisms are functions $G^n \to G^m$ of the form $(h_i)_i \mapsto (g_jh_{f(j)})_j$ where $f:[m] \to [n]$ (note that $n,m \in \{0,1\}$)
• whose set of equipped limits only consists of the terminal object $G^0 = 1$.

John Baez (Jun 28 2024 at 13:48):

What does it mean to say you've got a Lawvere theory whose only objects are $G^0$ and $G^1$? I must be misinterpreting you somehow.

Rémy Tuyéras (Jun 28 2024 at 13:50):

Thanks, edited to terminology

Eric M Downes (Jun 28 2024 at 14:55):

I conjecture that what we have been thinking of as the division $v:X^2\to G$ is actually a functor $V:{\sf Heap\to Tors}$. If instead of considering a ternary operator $t:X^3\to X$, you curry it, you get the signature $\tilde{t}:X^2\to X^X$ and it works exactly like a left group action. These diagrams make this clearest:

(I was using $Y$ for John's $H$; the set on which $G$ acts) Okay, gotta run, but I think the "transitive, free" part comes in when you need to establish equivalence. E.g. so that the effective action is not that of a subgroup of $G$

Rémy Tuyéras (Jun 28 2024 at 16:24):

Just to finish the details... in case it is useful...

• (if yes then) denoting $\mathsf{Mod}(T_G) \hookrightarrow [T_G,\mathbf{Set}]$ the reflective subcategory restricted to limit preserving functors, is the mapping $\mathcal{T}:G \mapsto \mathsf{Mod}(T_G)$ a functor $\mathsf{Grp} \to \mathbf{Cat}$ (or $\mathsf{Grp} ^{\mathsf{op}}\to \mathbf{Cat}$)?

Part 1 - Functorially mapping a group to a product sketch

To do this, I would define a functor $T:\mathsf{Grp} \to \mathsf{Cat}$ by

• mapping $G \mapsto T_G$ and
• sending a group morphism $u:G \to H$ to the morphism $T(u):T_G \to T_H$ in $\mathsf{Cat}$ defined by the following mapping rules:
  • on objects: $G^n \mapsto H^n$ for $n \in \{0,1\}$;
  • on arrows: $(\overline{g},f) \mapsto (u(\overline{g}),f)$ where $u(\overline{g}) = (u(g_j))_{j \in [n]}$ for $n \in \{0,1\}$.

This defines a morphism $T_G \to T_H$ in $\mathsf{Cat}$ because $u$ is a group homomorphism. This morphism also preserves the limit $G^0 = \mathbf{1}$ by definition.

Part 2 - Functorially mapping a group to a category of models (for a given product sketch)

Now, the functor $T:\mathsf{Grp} \to \mathsf{Cat}$ gives us a functor $\mathcal{T} = \mathsf{Mod}(T(-)):\mathsf{Grp}^{\mathsf{op}} \to \mathbf{Cat}$. The functor $\mathcal{T}$ is a subfunctor of the more obvious functor $[T(-),\mathsf{Set}]:\mathsf{Grp}^{\mathsf{op}} \to \mathbf{Cat}$ which maps a group $G$ to the functor category $[T(G),\mathsf{Set}]$.

Rémy Tuyéras (Jun 28 2024 at 16:49):

Part 3 - Checking the Grothendieck construction

Finally, the Grothendieck construction for the functor $\mathcal{T}:\mathsf{Grp}^{\mathsf{op}} \to \mathbf{Cat}$, usually denoted as $\int \mathcal{T}$, as defined as follows:

• objects are pairs $(G,H)$ consisting of a group $G$ and an object $H \in \mathcal{T}(G) = \mathsf{Mod}(T_G)$;
• arrows are pairs $(\phi,f)$ consisting of a group morphism $\phi:G \to G'$ and an arrow $f:H \to \mathcal{T}(\phi) \circ H'$ in $\mathcal{T}(G)$, hence a natural transformation $(f_k:H'(k)\Rightarrow H'(T(\phi)(k)) )_k$ giving the equation $f_k(gh) = \phi(g)f_k(h)$.

John Baez (Jun 28 2024 at 21:42):

I'm not clear on what the functor $\mathcal{T} \colon \mathsf{Grp}^{\rm{op}} \to \mathbf{Cat}$ is supposed to be, @Rémy Tuyéras. I get lost in all the details. Can you say in words what category it's supposed to map a group to?

Rémy Tuyéras (Jun 28 2024 at 21:57):

Ah right, I wrote $[T_G,\mathsf{Set}]$ but I meant $\mathsf{Mod}(T_G)$.

In a few words, $\mathcal{T}$ maps a group $G$ to the category of limit preserving functors $T_G \to \mathbf{Set}$ where $T_G$ is the product sketch I defined a few posts earlier

Nathan Corbyn (Jun 29 2024 at 11:09):

I’m confused why you need to go through finite limit sketches. If $\mathrm{Mod}(T_G)$ is supposed to be equivalent to $G\mathrm{-Tor}$, what stops you from defining the functor as mapping $G$ to $G\mathrm{-Tor}$ directly?

Nathan Corbyn (Jun 29 2024 at 11:11):

Or rather, if there’s a problem defining this as a (pseudo)functor explicitly (I think I can see it), there should still be a problem when you phrase it in terms of sketches

John Baez (Jun 29 2024 at 12:40):

Rémy Tuyéras said:

In a few words, $\mathcal{T}$ maps a group $G$ to the category of limit preserving functors $T_G \to \mathbf{Set}$ where $T_G$ is the product sketch I defined a few posts earlier.

Nathan seems to be saying it in fewer words: $\mathcal{T}$ maps a group $G$ to the category of $G$-torsors. Is that the goal here?

(I'm a bit confused by why you're using limit preserving functors out of a product sketch, instead of product preserving functors out of a product sketch or limit preserving functors out of a limit sketch.)

Rémy Tuyéras (Jun 29 2024 at 14:25):

John Baez said:

(I'm a bit confused by why you're using limit preserving functors out of a product sketch, instead of product preserving functors out of a product sketch or limit preserving functors out of a limit sketch.)

@John Baez This is an abuse of language on my part and my fault for trying to go too fast. I usually see a limit sketch as a pair $(C, L)$ where $C$ is a small category and $L$ is a set of cones in $C$. Here, I define a cone as a natural transformation of the form $\Delta_I(X) \Rightarrow F$ where:

• $I$ is a small category,
• $X$ is an object in $C$,
• $\Delta: C \to [I, C]$ is the obvious constant functor,
• $F: I \to C$ is a functor.

If all the small categories $I$ associated with the cones in $L$ are discrete categories, then I would tend to call $(C, L)$ a product sketch.

I define a model for a limit sketch as a functor $M: C \to \mathbf{Set}$ sending every cone $a:\Delta_I(X) \Rightarrow F$ in $L$ to a limit cone $M(a):\Delta_I(M(X)) \Rightarrow M \circ F$ in $\mathbf{Set}$.

While this is one definition, there are other common definitions for limit sketches. For instance, one might assume that the cones in $L$ must be limit cones in $C$. In that case, the models send the limit cones in $L$ to limit cones in $\mathsf{Set}$.

Based on my experience and interactions, I usually prefer to use the term limit-preserving to evoke the idea that these functors send the "equipped (synthetic) limits" to limits in $\mathbf{Set}$, avoiding potential confusion with the various meanings of "model". But in this case, that did not help me much with the clarity :upside_down:

(Also, I would love to hear how others define limit sketches so I can feel more comfortable with the terminology)

Rémy Tuyéras (Jun 29 2024 at 14:43):

Nathan Corbyn said:

I’m confused why you need to go through finite limit sketches. If $\mathrm{Mod}(T_G)$ is supposed to be equivalent to $G\mathrm{-Tor}$, what stops you from defining the functor as mapping $G$ to $G\mathrm{-Tor}$ directly?

That's a good point, and I see where you are coming from. But given why we are interested in $G$-torsors, I think it is useful to first understand them as models for limit sketches. Heaps can also be seen as models for limit sketches pretty straightforwardly (same with groups). Plus, if you check out the nLab page on [[heaps]], you will see that converting $G$-torsors to heaps and back mostly involves playing around with operations, hinting at deeper theoretical connections (as in theory for models).

If we are going to talk about the yoga of techniques that let us switch between them, I would rather get the big picture in the same language. That is why I lean towards limit sketches -- they would set the stage for understanding how groups, torsors, and heaps all fit together with their left adjoints.

Rémy Tuyéras (Jun 29 2024 at 14:55):

And maybe I should add that limit sketches are a very natural language to build those left adjoints, even giving us formulas, often useful for end, co-end and transfinite calculi.

For anyone interested in the topic, see Categories of continuous functors, I as a starting point.

Nathan Corbyn (Jun 29 2024 at 15:45):

I haven’t checked this carefully but I think it’s true that $G\mathrm{-Tors} \simeq \mathbf{B}G$. There’s a fully faithful functor from the delooping to torsors coming from taking $G$’s action on itself (Yoneda lemma) and (assuming torsors are assumed to be non-empty) then every tosor admits an isomorphism with $G$ as a set. The bit I haven’t checked is whether this isomorphism can always be made into an equivariant map.

In which case, we’re really talking about the functor $\mathbf{B} : \mathrm{Grp} \to \mathrm{Cat}$ sending each group to its delooping and how this compares to $\mathrm{Tors}$.

Todd Trimble (Jun 29 2024 at 18:26):

Question. Concretely as possible, what is the left adjoint of U:Grp→Heap?

And Oscar Cunningham proposed an answer.

The answer Oscar gave is the answer I would have given (without trying too hard to be concrete). In general, if you have monads $G, H$ on a category $\mathbf{C}$ with reflexive coequalizers, with a monad morphism $\phi: H \to G$, and $\mathbf{C}^G \to \mathbf{C}^H$ denotes the forgetful functor between the categories of algebras that is induced by the monad morphism, then the left adjoint to this forgetful functor takes an $H$-algebra $X$ to the coequalizer of a pair of arrows

$GHX \overset{\rho X}{\underset{G\lambda}{\rightrightarrows}} GX.$

This is just like the situation for writing down the tensor product of a right $R$-module $Y$ with a left $R$-module $X$, as a coequalizer:

$Y \otimes R \otimes X \rightrightarrows Y \otimes X \to Y \otimes_R X$.

Only this time, $H$ acts on $G$ on the right by a composite $GH \overset{G\phi}{\to} GG \overset{\mu}{\to} G$, and on the left on an $H$-algebra $X$ by the algebra structure $\lambda: HX \to X$.

Think for example of the way that the universal enveloping algebra of a Lie algebra works: by taking a certain quotient of the tensor algebra (which plays the role of $GX$). Oscar's proposal is working exactly the same way.

John Baez (Jun 29 2024 at 19:27):

Nathan Corbyn said:

I haven’t checked this carefully but I think it’s true that $G\mathrm{-Tors} \simeq \mathsf{B}G$

Yes, that's true. From one point of view the point of torsors is to take $\mathsf{B}G$, which has one object and one morphism for each element of $G$, and find an equivalent category with objects that are sets with extra structure - namely $G$-torsors - and structure-preserving maps between them. It gives a new and fruitful outlook.