To my brother Qr’istos and his wife Mar’ia on the occasion of their wedding.

Confiteor↩︎

I would like to apologize to everyone who has reached out to me enquiring with interest about this (many years old) work, aspects of which I have presented both in Cambridge and at the PSSL 101 (Leeds) in 2017. I am extremely fascinated by the Tannakian formalism and the current raison d’ être of my academic life is to explore how widely applicable it is - as I strongly believe that the formalism’s underlying mechanics are useful much more broadly than it has been recognized. While working on this more “fundamental” project, the work presented here was put aside, mainly because the “complete demystification" of the connection between Galois and Tannakian categories ended up being a task bigger than I initially anticipated. And even though I had naïvely hoped to explore every aspect of it alone before sharing my thoughts, the emotional weight that this attitude carries has now become unbearable; and I am thus opening Pandora’s box, in the hope that I will start making more material public. As a compromise, even though this article does present the main idea behind the connection, I will treat it as a”working” paper: I am planning to revisit it with additions and more details, as the state of it is by no means complete.

The worlds of Tannakian and Galois categories are beautiful gardens bearing fruits from many different areas of mathematics, and I have tried to demonstrate this by using rather diverse language here - the first part is written with Algebraic Geometers in mind while the second is written with Category Theorists in mind. In fact I see the exploration of the connection between Galois and Tannakian categories as a great cooperation opportunity between these communities: in my opinion insight from both sides is necessary in order to harvest the full potential of this relationship. I welcome any questions, suggestions and feedback on this work, but most importantly (the reason for deciding to publish this in a working paper state) I would like to invite anyone who has thoughts and insights, either from a geometric or a categorical perspective, to write about them, as I really believe there are aspects that I have missed but also fruitful directions that deserve to be developed independently. Studying this connection seriously is a few decades overdue, and this has to be a collaborative effort.

Acknowledgments↩︎

Firstly I would like to thank Martin Hyland for all his support: for accepting my suggestion to study the connection between Galois and Tannakian categories as a research project in the first place, for allowing me complete freedom in my work, for not losing hope in me regardless of my secretive attitude, and for his numerous attempts at convincing me to write my thoughts down in detail. I would also like to thank Pierre Deligne who, through his works and our correspondence, has taught me how to use the Tannakian formalism in a “working” way - the interpretation of my adjunction using “Tannakian Algebraic Geometry” is a result of his guidance. Finally, I would like to thank Peter Johnstone for his interest and encouragement for this work in the early stages.

1 ↩︎

In this first part I will use the language of gerbes and Deligne’s theory of “Algebraic Geometry internal in a Tannakian category” to describe the connection between Tannakian and Galois categories “both internally and externally” - what this means will be made clear in the body of the text.

1.0.1 The general picture↩︎

1.0.1.1 The \(1\)-dimensional situation

Let me start by discussing the \(1\)-dimensional situation that the aforementioned procedure “categorifies”. Let \(k\) be a field and consider the inclusion \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{AlgGrpSch}}}_k\\ \mathop{\mathrm{\mathbf{\acute{E}tGrpSch}}}_k\arrow[u,hook, "\iota"] \end{tikzcd}\] of the full subcategory of étale (algebraic) group \(k\)-schemes in the category of (affine) algebraic group \(k\)-schemes. Every algebraic group \(k\)-scheme \(G\) induces the short exact sequence of algebraic group \(k\)-schemes \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/mueibjwv.png}\label{bshdemry}\end{figure}\tag{1}\] where

• the (finite) étale group \(k\)-scheme \(\pi_0(G)\) of connected components is defined as \[\pi_0(G):=\mathop{\mathrm{Spec}}L,\] with \(L\) the largest separable \(k\)-subalgebra of the finitely generated commutative Hopf \(k\)-algebra \(\mathcal{O}(G)\), and with the map \(q_G\), induced by the inclusion \[L\subseteq\mathcal{O}(G),\] being a surjective map of (affine) group \(k\)-schemes courtesy of the fact that \(\mathcal{O}(G)\) is faithfully flat over \(L\),

• and the connected component or identity component algebraic group \(k\)-scheme \(G^0\), defined as the kernel of \(q_G\), is connected.

This construction of \(\pi_0(G)\) is functorial, and the induced functor \(\pi_0\) is a left adjoint to the inclusion \(\iota\), with every \(q_G\) being an epimorphism (a surjective map of affine group \(k\)-schemes is easily seen to be an epimorphism in \(\mathop{\mathrm{\mathbf{AlgGrpSch}}}_k\)), realizing \(\mathop{\mathrm{\mathbf{\acute{E}tGrpSch}}}_k\) as a full, epi-reflective subcategory \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{AlgGrpSch}}}_k\arrow[d,shift left=1.4ex, "\pi_0"]\\ \mathop{\mathrm{\mathbf{\acute{E}tGrpSch}}}_k\arrow[u,hook,shift left=1.4ex, "\iota", "\vdash"'] \end{tikzcd}\] of \(\mathop{\mathrm{\mathbf{AlgGrpSch}}}_k\).

The next observation is that this reflection \(\mathop{\mathrm{\mathbf{Pro}}}\)-completes: indeed, every affine group \(k\)-scheme \(G\) can be written as a cofiltered limit \[G\cong\varprojlim_i G_i\] of algebraic \(k\)-group schemes \(G_i\), with each \(G_i=\mathop{\mathrm{Spec}}\mathcal{O}(G)_i\) corresponding to the finitely generated commutative Hopf \(k\)-subalgebras \(\mathcal{O}(G)_i\subseteq \mathcal{O}(G)\) realizing \[\mathcal{O}(G)_i\cong\varinjlim_i \mathcal{O}(G)_i,\] as the filtered union of the \(\mathcal{O}(G)_i\). In fact, \[\mathop{\mathrm{\mathbf{CHopf}}}_k\simeq\mathop{\mathrm{\mathbf{Ind}}}\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\] and \[\mathop{\mathrm{\mathbf{AffGrpSch}}}_k\simeq \mathop{\mathrm{\mathbf{Pro}}}\mathop{\mathrm{\mathbf{AlgGrpSch}}}_k.\] Taking the group of connected components of every \(G_i\) induces a canonical morphism \[q_G:G\cong\varprojlim_i G_i\to\varprojlim_i \pi_0(G_i)=:\pi_0(G),\] to a pro- (finite) étale group \(k\)-scheme whose kernel identifies with \[\ker q_G\cong \varprojlim_i G_i^0,\] and it is connected.

This procedure of “taking the maximal pro-étale quotient of an affine \(k\)-group scheme” induces an epi-reflection \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{AffGrpSch}}}_k\arrow[d,shift left=1.4ex, "\pi_0"]\\ \mathop{\mathrm{\mathbf{AffGrpSch}}}_k^{\mathop{\mathrm{pro\acute{e}t}}}\arrow[u,hook,shift left=1.4ex, "\iota", "\vdash"'] \end{tikzcd}\] where \(\iota\) is now the inclusion in \(\mathop{\mathrm{\mathbf{AffGrpSch}}}_k\) of the full subcategory \(\mathop{\mathrm{\mathbf{AffGrpSch}}}_k^{\mathop{\mathrm{pro\acute{e}t}}}\) consisting of those affine \(k\)-group schemes that can be written as cofiltered limits of étale group \(k\)-schemes.

Let’s see now how the situation “externalizes” by picking a separable closure of \(k\): let \(\overline{k}\) be one such separable closure of \(k\). As it can be gathered from the upcoming Section 3.2, the assignment \[G\mapsto G(\overline{k}),\] sending a finite étale \(k\)-scheme (or finite étale cover, in the terminology of Section 3.2) to its (“rational”) \(\overline{k}\)-points induces an equivalence \[\mathop{\mathrm{\mathbf{F\acute{E}t}}}_k\simeq \mathop{\mathrm{Cont}}_f(\mathop{\mathrm{\mathbf{Gal}}}(\overline{k}/k)),\] between the category of finite étale \(k\)-schemes and the category of discrete finite sets endowed with a continuous action of the profinite (Stone) absolute Galois group of \(k\). Choosing now \(k\) to be separably closed kills the absolute Galois group, \(\mathop{\mathrm{Cont}}_f(\mathop{\mathrm{\mathbf{Gal}}}(\overline{k}/k))\) identifies with the category \(\mathop{\mathrm{\mathbf{Set}}}_f\) of finite sets and by passing to the group objects we end up with an equivalence \[\begin{align} \mathop{\mathrm{\mathbf{\acute{E}tGrpSch}}}_k&\to&\mathop{\mathrm{\mathbf{Grp}}}_f\\ G&\mapsto& G(k). \end{align}\] By composing with \(\pi_0\), we get a functor \[\begin{align} \mathop{\mathrm{\mathbf{AlgGrpSch}}}_k&\to&\mathop{\mathrm{\mathbf{Grp}}}_f\\ G&\mapsto& \pi_0(-)_k:=\pi_0(-)(k). \end{align}\] which is left adjoint to the fully faithful functor \[\begin{align} \mathop{\mathrm{\mathbf{Grp}}}_f&\to&\mathop{\mathrm{\mathbf{AlgGrpSch}}}_k\\ G&\mapsto& k^G, \end{align}\] where \(k^G\) is the coproduct of \(|G|\)-many copies of \(\mathop{\mathrm{Spec}}k\); i.e. we have a reflection \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{AlgGrpSch}}}_k\arrow[d,shift left=1.4ex, "\pi_0(-)_k"]\\ \mathop{\mathrm{\mathbf{Grp}}}_f\arrow[u,hook,shift left=1.4ex, "k^{(-)}", "\vdash"'] \end{tikzcd}\] which \(\mathop{\mathrm{\mathbf{Pro}}}\)-completes to give an adjunction (the “external” incarnation of the reflection \(\pi_0\dashv \iota\) of \(\mathop{\mathrm{\mathbf{AffGrpSch}}}_k^{\mathop{\mathrm{pro\acute{e}t}}}\) in \(\mathop{\mathrm{\mathbf{AffGrpSch}}}_k\) for \(k\) separably closed), \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{AffGrpSch}}}_k\arrow[d,shift left=1.4ex, "\mathop{\mathrm{\mathbf{Pro}}}\pi_0(-)_k"]\\ \mathop{\mathrm{\mathbf{StoneGrp}}}\arrow[u,hook,shift left=1.4ex, "I", "\vdash"'] \end{tikzcd}\] where I have replaced (as a harbinger to a connection with the Pierce spectrum I will mention later) the expected functor \(\mathop{\mathrm{\mathbf{Pro}}}k^{(-)}\) with the functor \(I\) defined as follows, \[\begin{align} I:\mathop{\mathrm{\mathbf{StoneGrp}}}&\to&\mathop{\mathrm{\mathbf{AffGrpSch}}}_k\\ \pi&\mapsto& \mathop{\mathrm{Spec}}\mathop{\mathrm{Cont}}(\pi,k_{\mathop{\mathrm{disc}}}), \end{align}\] with \(\mathop{\mathrm{Cont}}(\pi,k_{\mathop{\mathrm{disc}}})\) being the commutative Hopf \(k\)-algebra of continuous functions from \(\pi\) to the field \(k\) endowed with the discrete topology.

1.0.1.2 The \(2\)-dimensional situation

Now let’s look at general Tannakian categories over a field \(k\) and Galois categories collectively. Write \(\mathop{\mathrm{\mathbf{Tan}}}_k\) and \(\mathop{\mathrm{\mathbf{Gal}}}\) for their \(2\)-categories respectively.

I would like the reader to think of the situation here as similar in spirit to its \(1\)-dimensional counterpart discussed in the previous subsection. There, I argued that for \(k\) separably closed, one can associate to a Stone group \(\pi\) an affine (pro-étale) group \(k\)-scheme \(I(\pi)\) and from it one can go back to (“recover”) \(\pi\) canonically (cf. Remark 3). The point is that the same holds, in an appropriately \(2\)-dimensional way, for Galois and Tannakian categories: for any field \(k\), one can associate, uniquely, a Galois category \(\mathcal{C}\) (without any chosen fibre functor) a (neutral) Tannakian category \(\mathcal{C}_{\mathop{\mathrm{lin}}}\), and when \(k\) is separably closed, one can in fact recover \(\mathcal{C}\) from \(\mathcal{C}_{\mathop{\mathrm{lin}}}\). In this case, \(\mathcal{C}\) and \(\mathcal{C}_{\mathop{\mathrm{lin}}}\) have “the same” fibre functors.

Now let me describe how the (co-²) reflection is actually constructed - there are, again, internal and external ways to look at this. To understand the effect of the \(2\)-coreflection internally - again I use this phrase just to signify that everything happens within the \(2\)-category \(\mathop{\mathrm{\mathbf{Tan}}}_k\) - let \(\mathbb{T}\) be a Tannakian category over \(k\) and write \(\pi(\mathbb{T})\) for the fundamental group of \(\mathbb{T}\), as constructed in Section 8 of [2]; it is an affine group \(\mathbb{T}\)-scheme, \[\pi(\mathbb{T})\in\mathop{\mathrm{\mathbf{AffGrp}}}_{\mathbb{T}}:=\mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{CMon}}}(\mathop{\mathrm{\mathbf{Ind}}}\mathbb{T})^{\mathop{\mathrm{op}}})\] which acts on every object of \(\mathbb{T}\). Now notice that the construction of the \(\pi_0(G)\) of an affine group \(k\)-scheme \(G\) is compatible with base change, and thus we can form the internal incarnation \[\pi_0(\pi(\mathbb{T}))\] of \(\pi_0\) (see Exemple 8.14 of [2]), by performing the construction of the previous subsection on the “realized” affine group \(k\)-schemes \[\pi_0(\pi(\mathbb{T}))_{\omega}\] for every fibre functor \(\omega\). This procedure induces a faithfully flat morphism \[q_{\pi(\mathbb{T})}:\pi(\mathbb{T})\to \pi_0(\pi(\mathbb{T}))\] in \(\mathop{\mathrm{\mathbf{AffGrp}}}_{\mathbb{T}}\). The kernel \[\pi(\mathbb{T})^0:=\ker q_{\pi(\mathbb{T})}\] is a normal affine group sub-\(\mathbb{T}\)-scheme of \(\pi(\mathbb{T})\), and in fact we get a short exact sequence of affine group \(\mathbb{T}\)-schemes \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/tqchyfeb.png}\label{xjpeatco}\end{figure}\tag{2}\] The normal group sub-\(\mathbb{T}\)-scheme \(\pi(\mathbb{T})_0\) of \(\pi(\mathbb{T})\) induces a Tannakian subcategory \[\Pi(\mathbb{T})\subseteq\mathbb{T}\] generated by the objects of \(\mathbb{T}\) on which the connected component \(\pi(\mathbb{T})^0\) acts trivially (cf. [3], 6.6), which, from the gerbe perspective, corresponds to an fpqc gerbe whose band is a profinite group \(k\)-scheme.

Assuming \(k\) to be separably closed, i.e. entering the realm in which general Galois categories become co-reflective in Tannakian categories, the rôle of the “internal” \(1\)-dimensional reflector \(\pi_0\) is played at the \(2\)-dimensional level by the (co-) reflective association \[\mathbb{T}\mapsto \Pi(\mathbb{T}).\] In this case, the band of the Tannakian gerbe of \(\Pi(\mathbb{T})\) corresponds to a classical profinite (Stone) group, and the gerbe becomes a gerbe with a Stone band, which in turn corresponds to a Galois category \(\mathbf{Gal}_{\mathbb{T}}\). The rôle of the external \(1\)-dimensional reflector \(\pi_0(k)\) is played, at the \(2\)-dimensional level, by Carboni’s construction: the Galois category \(\mathbf{Gal}_{\mathbb{T}}\) can be obtained from the Tannakian category \(\mathbb{T}\) as the Boolean pretopos that Carboni proves to be associable to any compact closed additive category with equalizer, by taking commutative separable algebras in \(\mathbb{T}\).

2 ↩︎

In Part II that follows I elaborate further on the idea discussed above, by demonstrating how the (co-) reflection works, in more detail, in the case where neutralized versions of Tannakian and Galois categories are considered. Notice that this is morally unacceptable, as interesting Tannakian categories, like those of motives, come with many available “realization” fibre functors, and choosing one is criminal; however doing so, as I will explain below, essentially eliminates the effects of \(2\)-dimensional category theory, allowing me to pass the general idea in a simpler categorical environment. I give all the definitions and results in detail.

2.0.1 Galois categories↩︎

2.0.1.1 Definitions

The main reference for Galois categories is [4]. The definition that follows is not Grothendieck’s original axiomatic characterization, but it is equivalent to it; it consists of a more compact set of axioms than those appearing in [4].

In this work I will be studying (neutralized) Galois categories collectively so let’s see what a \(1\)-morphism between them should be.

Next let’s see what \(2\)-morphisms are.

We must immediately notice that this \(2\)-category is not exactly a \(2\)-category: it is, rather a “setoid-enriched category”, i.e. \(\mathop{\mathrm{\mathbf{Gal}}}_{*}\) is naturally \(2\)-equivalent to a \(1\)-category: one can check that \(2\)-morphisms are isomorphisms and automorphisms of \(1\)-morphisms are the identity. Indeed, notice that on components, for every \(C\in \mathop{\mathrm{ob}}\mathcal{C}\), we have by the Godement rule that \[\begin{align} \label{eqq} \alpha_C=(\alpha'\cdot \tau)_C=\alpha'_C\circ(\iota_{p'}\underset{2}{\circ} \tau)_C=\alpha'_C\circ \mathop{\mathrm{id}}_{pF'C}\circ p'(\tau_C)=\alpha'_C\circ p'(\tau_C), \end{align}\tag{5}\] implying that, since both \(\alpha_C\) and \(\alpha'_C\) are isomorphisms, \(p'(\tau_C)\) must be an isomorphism too. But it then immediately follows from requirement \(\text{(GAL5)}\) that \(\tau_C\) is also an isomorphism for every \(C\in \mathop{\mathrm{ob}}\mathcal{C}\), so \(2\)-morphisms are indeed isomorphisms and we are half-way there. We need to also show that automorphisms of \(1\)-morphisms are the identity \(2\)-morphism. But this follows easily by noticing that \((\ref{eqq})\) becomes in this case \[\alpha_C=\alpha_C\circ p'(\tau_C),\] which in turn implies \[p'(\tau_C)=\mathop{\mathrm{id}}_{p'C}\] since \(\alpha_C\) is an isomorphism. It then suffices to notice that a fibre functor is faithful and the identity already gives \(p'(\mathop{\mathrm{id}}_C)=\mathop{\mathrm{id}}_{p'C}\). Thus \(\tau_C\) must indeed be the identity itself.

2.0.1.2 Classification of neutralized Galois categories

For a profinite group \(\pi\), denote by \[\mathop{\mathrm{Cont}}_f(\pi)\] the category of discrete finite sets endowed with a continuous (left) action of \(\pi\). It is a Galois category admitting the forgetful functor \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/loyiftuc.png}\label{xutzdpkl}\end{figure}\tag{6}\] to the category of finite sets as a fibre functor. It turns out that all (neutralized) Galois categories are of this form. To see this we need the following definition.

This fundamental group is a profinite Stone group when topologized as a closed subgroup \[\pi_1^{\mathop{\mathrm{Gal}}}(\mathcal{C},p)\subseteq\prod_{C\in\mathop{\mathrm{ob}}\mathcal{C}} \mathop{\mathrm{Aut}}(FC),\] of the product group above, which is itself a profinite group when endowed with the product topology (each of the finite groups \(\mathop{\mathrm{Aut}}(FC)\) is given the discrete topology), as an arbitrary product of profinite groups is profinite and discrete finite groups are profinite.

Grothendieck’s work on Galois categories has the following two equally important aspects:

• Reconstruction: For any profinite group we have \(\pi\cong\pi_1^{\mathop{\mathrm{Gal}}}(\mathop{\mathrm{Cont}}_f(\pi),U)\).

• Recognition: A neutralized Galois category \((\mathcal{C},p)\) factorizes through an equivalence \[\begin{tikzcd}[row sep=large, column sep=large] \mathcal{C}\arrow[r, "\simeq"]\arrow[d, "p"']& \mathop{\mathrm{Cont}}_f\left(\pi_1^{\mathop{\mathrm{Gal}}}(\mathcal{C},p)\right) \arrow[ld, "U"]\\ \mathop{\mathrm{\mathbf{Set}}}_f \end{tikzcd}\]

These constructions are (\(2\)-) functorial and we have the following result.

2.0.2 Carboni’s separable algebras↩︎

Every field \(k\) defines its Galois category \(\mathop{\mathrm{\mathbf{F\acute{E}t}}}_k\), the category of “finite étale covers over \(\mathop{\mathrm{Spec}}k\)”, and to get a fibre functor on it one needs to pick a separable closure of \(k\) (which is where the virtue of choosing \(k\) to be separably closed in this article comes in). This geometrically defined category admits a description using the algebraic notion of “separable” or “étale” \(k\)-algebra, which is precisely the notion on which Carboni builds his generalized notion of commutative separable algebra in a symmetric monoidal category that I will be using to pass from Tannakian to Galois categories. In order to give meaning to the statements I made above, I would like to discuss the notion of finite étale schemes in some detail and in particular explain their algebraic side. Thus this section starts with a brief introduction to the relevant scheme-theoretic notions.

2.0.2.1 Basics of Scheme Theory

Let’s start with defining some fundamental scheme-theoretic notions.

• A ringed space is a pair \((V,\mathcal{O}_V)\) consisting of a topological space \(V\) and a sheaf of rings \(\mathcal{O}_V\) called the structure sheaf.

• A locally ringed space is a ringed space \((V,\mathcal{O}_V)\) if the stalks \(\mathcal{O}_{V,v}\) of \(\mathcal{O}_V\) at every \(v\in V\) are local rings. A morphism of locally ringed spaces \((V,\mathcal{O}_V)\to(W,\mathcal{O}_W)\) is a pair \((f,f^{\sharp})\) consisting of

  • a continuous morphism \(f:V\to W\) of topological spaces,

  • a morphism \(f^{\sharp}:\mathcal{O}_W\to f_{\ast}\mathcal{O}_V\) of sheaves of rings,

  • such that for any \(v\in V\) the homomorphism of rings on the stalks \[f^{\sharp}_v:\mathcal{O}_{W,f(v)}\to \mathcal{O}_{V,v}\] is a homomorphism of local rings.

• For any commutative ring \(R\) one can define a topology on the set \(\mathop{\mathrm{Spec}}R\) of prime ideals of \(R\), called the Zariski topology, having a basis of open sets given, for every \(r\in R\), by the sets \[D(r):=\lbrace P\;:\;\text{P prime with r\notin P} \rbrace,\] and which admits a structure sheaf \(\mathcal{O}_{\mathop{\mathrm{Spec}}R}\), making \((\mathop{\mathrm{Spec}}R, \mathcal{O}_{\mathop{\mathrm{Spec}}R})\) into a locally ringed space called the (prime) spectrum of \(R\). Notice that spectra of fields are just points as the only prime ideal of a field \(k\) is the trivial ideal so \(\mathop{\mathrm{Spec}}k=\lbrace (0)\rbrace\).

• An affine scheme is a locally ringed space \((V,\mathcal{O}_V)\) isomorphic to the ringed space \((\mathop{\mathrm{Spec}}R, \mathcal{O}_{\mathop{\mathrm{Spec}}R})\) for some commutative ring \(R\). A morphism of affine schemes is a morphism of locally ringed spaces. The category of affine schemes will be denoted by \(\mathop{\mathrm{\mathbf{Aff}}}\).

• A scheme is a locally ringed space that admits an open covering by affine schemes. A morphism of schemes is a morphism of locally ringed spaces. The category of schemes will be denoted by \(\mathop{\mathrm{\mathbf{Sch}}}\).

In fact the construction of the (affine) scheme \(\mathop{\mathrm{Spec}}R\) out of a commutative ring \(R\), assembles into a fully faithful functor \[\begin{align} \mathop{\mathrm{Spec}}:\mathop{\mathrm{\mathbf{CRing}}}^{\mathop{\mathrm{op}}}&\to &\mathop{\mathrm{\mathbf{Sch}}}\\ R&\mapsto & (\mathop{\mathrm{Spec}}R,\mathcal{O}_{\mathop{\mathrm{Spec}}R}). \end{align}\] Taking global sections of the structure sheaves of schemes (or taking the coordinate ring \(\mathcal{O}(V)\) of a scheme \((V,\mathcal{O}_V)\), terminology inherited from the situation generalized here from the level of affine varieties) induces a functor \[\begin{align} \mathcal{O}:\mathop{\mathrm{\mathbf{Sch}}}&\to &\mathop{\mathrm{\mathbf{CRing}}}^{\mathop{\mathrm{op}}}\\ (V,\mathcal{O}_V) &\mapsto & \Gamma_V(\mathcal{O}_V)=\mathcal{O}_V(V)=:\mathcal{O}(V), \end{align}\] which is left adjoint to the functor \(\mathop{\mathrm{Spec}}\). In this way \(\mathop{\mathrm{\mathbf{CRing}}}^{\mathop{\mathrm{op}}}\) becomes a reflective subcategory of \(\mathop{\mathrm{\mathbf{Sch}}}\), \[\begin{tikzcd}[row sep=large, column sep=large] \mathop{\mathrm{\mathbf{Sch}}}\arrow[d, shift left=1ex, "\mathcal{O}"]\\ \mathop{\mathrm{\mathbf{CRing}}}^{\mathop{\mathrm{op}}} \arrow[u, hook, shift left=1ex, "\mathop{\mathrm{Spec}}", "\vdash"'] \end{tikzcd}\] which realizes \(\mathop{\mathrm{\mathbf{CRing}}}^{\mathop{\mathrm{op}}}\) as \[\mathop{\mathrm{\mathbf{CRing}}}^{\mathop{\mathrm{op}}}\simeq \mathop{\mathrm{\mathbf{Aff}}},\] the full subcategory of \(\mathop{\mathrm{\mathbf{Sch}}}\) consisting of affine schemes.

Now let us recall Grothendieck’s relative point of view, in which the focus is not just on schemes in isolation, but rather, schemes relative to some “base scheme”.

Every ring admits a unique morphism from the ring of integers \(\mathbb{Z}\), and every scheme admits a unique morphism to the scheme \(\mathop{\mathrm{Spec}}\mathbb{Z}\), courtesy of the structure adjunction bijection \[\relax_{\mathop{\mathrm{\mathbf{Sch}}}}(X,\mathop{\mathrm{Spec}}R)\cong\relax_{\mathop{\mathrm{\mathbf{CRing}}}}(R, \mathcal{O}(X)),\] allowing us to think of all schemes as schemes over \(\mathbb{Z}\). We want to obtain a “relative to a base scheme \(S\)” version of the structure adjunction. For this, we need to be able to “base-change”.

Now let us fix a ring \(R\). Since \(\mathop{\mathrm{\mathbf{Sch}}}\) has pullbacks, we can look at the “over \(R\)” version of the structure adjunction.

2.0.2.2 Finite étale covers and classical separable \(k\)-algebras

The motivating example of a Galois category for Grothendieck was the category \(\mathop{\mathrm{\mathbf{F\acute{E}t}}}_X\) of finite étale covers of a connected scheme \(X\), i.e. the full subcategory \[\mathop{\mathrm{\mathbf{F\acute{E}t}}}_X\subset \mathop{\mathrm{\mathbf{Sch}}}/X\] consisiting of the schemes over \(X\) which are “finite and étale⁴”. For the general definitions see Exposé I of [4].

For every choice of a geometric point \(\overline{x}:\mathop{\mathrm{Spec}}\Omega\to X\) of \(X\), where \(\Omega\) is a separably closed field, and for any finite étale \(X\)-scheme \(f:S\to X\), consider the pullback \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/unrgwpys.png}\label{rhlbfdmo}\end{figure}\tag{7}\] and denote by \[\phi_{\overline{x}}(S):=|S\times_X \mathop{\mathrm{Spec}}\Omega|\] the underlying set of the finite étale scheme \(S\times_X \mathop{\mathrm{Spec}}\Omega\) over \(\mathop{\mathrm{Spec}}\Omega\), which is finite by Remark 20. In this way we get a fibre functor \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ykfpzwxq.png}\label{zlgyhesq}\end{figure}\tag{8}\] on \(\mathop{\mathrm{\mathbf{F\acute{E}t}}}_X\) sending a finite étale cover \(S\) of \(X\) to \(\phi_{\overline{x}}(S)\).

Let’s now discuss the case \(X=\mathop{\mathrm{Spec}}k\) in more detail. In this case, a finite étale scheme \(S\to\mathop{\mathrm{Spec}}k\) takes the form \[S=\mathop{\mathrm{Spec}}A,\] where \(A\) is a “(finite) commutative separable \(k\)-algebra”. Let’s recall the definition of these objects, as it is on this algebraic side of the theory that Carboni builds his monoidal notion.

I will denote by \[\mathop{\mathrm{\mathbf{FCSep}}}_k\] the full subcategory of \(\mathop{\mathrm{\mathbf{CAlg}}}_k\) on the finite⁵ separable \(k\)-algebras. The structure adjunction induces an equivalence \[\begin{align} \label{fetsep} \mathop{\mathrm{\mathbf{F\acute{E}t}}}_{\mathop{\mathrm{Spec}}k}\simeq \mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}}. \end{align}\tag{9}\] Notice the following standard characterization of separability, which is the algebraic parallel of Remark 20.

Thus when \(k\) is separably closed, every finite commutative separable algebra over \(k\) is of the form \(k^n\), inducing the expected contravariant equivalence \[\mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}}\simeq \mathop{\mathrm{\mathbf{Set}}}_f.\] Under equivalence \((\ref{fetsep})\), the fibre functor \(p:\mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}}\to\mathop{\mathrm{\mathbf{Set}}}_f\) can be identified with the functor \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/nxfligjq.png}\label{duitxbjm}\end{figure}\tag{10}\]

The Galois fundamental group of this neutralized Galois category based at \(p\) identifies as \[\pi_1^{\mathop{\mathrm{Gal}}}(\mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}},p)\cong \mathop{\mathrm{\mathbf{Gal}}}(\overline{k}/k),\] the absolute Galois group of \(k\) (which is profinite with the Krull topology). This fibre functor thus induces, by Theorem \(\ref{thmgal}\), an equivalence \[\begin{align} \label{contgal} \mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}}\simeq \mathop{\mathrm{Cont}}_f(\mathop{\mathrm{\mathbf{Gal}}}(\overline{k}/k)) \end{align}\tag{11}\] for any choice \(\overline{k}\) of a separable closure of \(k\).

2.0.2.3 Carboni’s Commutative Separable Algebras in a monoidal category

Let me give the definition appearing in [1] right away; a discussion of how to recover the classical separable algebras appearing in Grothendieck’s reformulation of Galois theory will follow.

This definition is inspired by the following definition, that Carboni attributes to DeMeyer and Ingraham citing [6].

One can then use the fact that for any DeMI-separable algebra \(A\) over \(R\) there exists a left \(A\otimes_R A^{op}\)-module (epi-) morphism of left \(A\otimes A^{\mathop{\mathrm{op}}}\)-modules \[\begin{align} m:A\otimes_R A^{\mathop{\mathrm{op}}}&\to& A\\ \sum_i a_i\otimes a_i'&\mapsto&\sum_i a_i a_i', \end{align}\] called the augmentation map of \(A\), for which the short exact sequence \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/swjteuqp.png}\label{bvtofcpy}\end{figure}\tag{13}\] splits (i.e. the augmentation map \(m\) admits a section), to verify that commutative DeMI-separable algebras over \(R\) can be identified, in the symmetric monoidal category \(\mathop{\mathrm{\mathbf{Proj}}}_R\) of projective modules over a commutative ring \(R\), with those commutative algebras \((A,\mu,\eta)\) admitting also a cocommutative coalgebra structure \((A,\delta,\epsilon)\), which satisfy precisely axioms (U) and (D) of Definition \(\ref{defcsepabs}\).

For the purposes of this article, Carboni’s terminology is quite satisfactory, as his notion of separabe algebra is used in a Galois-theoretic context. However, this notion appears in modern literature in other contexts, the following being two of them which I consider useful to keep in mind.

• Denoting by \(\mathop{\mathrm{\mathbb{F}}}\) the PROP for commutative monoids and by \(\mathop{\mathrm{\mathbb{F}}}^{\mathop{\mathrm{op}}}\) the PROP for commutative comonoids, one can view commutative separable algebras in \(\mathop{\mathrm{\mathcal{V}}}\) as the algebras for the composite prop \[\mathop{\mathrm{\mathbb{F}}}^{\mathop{\mathrm{op}}}\otimes_{\mathbb{P}}\mathop{\mathrm{\mathbb{F}}}\] as in Example 5.4 of [7].

• Carboni’s notion of a commutative separable monoid also coincides with the notion of a special commutative Frobenius monoid appearing in more modern literature. In my opinion saying “special” to refer to a commutative Frobenius monoid satisfying axiom (U) of Definition \(\ref{defcsepabs}\) is not ideal - perhaps “split commutative Frobenius monoid” is better terminology, because of the splitting of the augmentation-induced short exact sequence at the level of DeMI-separable algebras.

I still haven’t explained how to assemble commutative separable monoids in \(\mathop{\mathrm{\mathcal{V}}}\) into a category and this is on purpose - it requires a bit of attention. Indeed one might expect that the morphisms between commutative separable monoids should be those which are both monoid and comonoid homomorphisms. However this is not a good choice: in this case such morphisms are automatically invertible and we end up with a groupoid of them.

Instead, we want to use only one kind of morphisms, either monoid or comonoid homomorphisms.

Carboni proves that these two categories are dual to each other.

Let’s define next the categories on which Carboni’s construction will be applied.

2.0.3 Tannakian Categories↩︎

The main references for Tannakian categories are [2], [8] and [9].

2.0.3.1 Definitions

Throughout this section \(k\) will denote a field.

The notion of a fibre functor is more complicated in the case of Tannakian categories than in the case of Galois categories.

We can now give the definition of a Tannakian category.

As in the case of Galois categories, we will be studying neutralized \(k\)-Tannakian categories collectively so and let’s define \(1\)-morphisms.

Next let’s see what \(2\)-morphisms are.

2.0.3.2 Classification of neutralized Tannakian categories

2.0.3.2.1 Affine groups over a field \(k\)

There are three equivalent ways to define the “concrete” objects corresponding to neutralized Tannakian categories over a field \(k\): the “geometric”, “functorial” and “algebraic” languages of affine group schemes over \(k\), affine groups over \(k\) and commutative Hopf algebras over \(k\), respectively. Let’s define these main actors of the theory right away.

• Affine group \(k\)-schemes: Notice that the category \(\mathop{\mathrm{\mathbf{Aff}}}_k\) is cartesian, with the product of affine \(k\)-schemes \(X=\mathop{\mathrm{Spec}}A\) and \(Y=\mathop{\mathrm{Spec}}B\) given by the pullback \[\begin{tikzcd}[row sep=large, column sep=large] X\times_{\mathop{\mathrm{Spec}}k} Y\arrow[d, dotted] \arrow[r, dotted]&X=\mathop{\mathrm{Spec}}A\arrow[d]\\ Y=\mathop{\mathrm{Spec}}B\arrow[r]&\mathop{\mathrm{Spec}}k \end{tikzcd}\] where \[X\times_{\mathop{\mathrm{Spec}}k} Y=\mathop{\mathrm{Spec}}(A\otimes_k B).\] An affine group scheme over \(k\) or affine group \(k\)-scheme is then a group object in the category of affine \(k\)-schemes and their category will be denoted by \[\mathop{\mathrm{\mathbf{AffGrpSch}}}_k:=\mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{Aff}}}_k).\]

• Affine \(k\)-groups: An affine group over \(k\) or affine \(k\)-group is a functor \(G:\mathop{\mathrm{\mathbf{CAlg}}}_k\to\mathop{\mathrm{\mathbf{Grp}}}\), which when composed with the forgetful functor \(U:\mathop{\mathrm{\mathbf{Grp}}}\to\mathop{\mathrm{\mathbf{Set}}}\) becomes representable by a \(k\)-algebra denoted by \(\mathcal{O}(G)\) and called the coordinate algebra of \(G\) \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/wgbphcfi.png}\label{wxqkbvta}\end{figure}\tag{16}\] A morphism of affine \(k\)-groups \(G\to H\) is a natural transformation \(G\Rightarrow H\) and the resulting category of affine \(k\)-groups will be denoted by \[\mathop{\mathrm{\mathbf{AffGrp}}}_k:=\left[\mathop{\mathrm{\mathbf{CAlg}}}_k,\mathop{\mathrm{\mathbf{Grp}}}\right]_{\text{rep}}\] This definition is equivalent to that of an affine \(k\)-group scheme as a consequence of the behaviour of models (group objects in this case) in presheaf toposes, which is accessible upon embedding \(\mathop{\mathrm{\mathbf{Aff}}}_k\simeq \mathop{\mathrm{\mathbf{CAlg}}}_k^{\mathop{\mathrm{op}}}\) in copresheaves on \(\mathop{\mathrm{\mathbf{CAlg}}}_k\) via Yoneda and using the fact that this Yoneda embedding preserves finite products and thus group objects.

• Commutative Hopf \(k\)-algebras: Reversing the point of view in the last statement, i.e. by thinking of affine \(k\)-groups as group objects in representable copresheaves on \(\mathop{\mathrm{\mathbf{CAlg}}}_k\), notice that

  • the group object multiplication \[\text{mult}:\mathop{\mathrm{\mathbf{CAlg}}}_k(\mathcal{O}(G),-)\times\mathop{\mathrm{\mathbf{CAlg}}}_k(\mathcal{O}(G),-)\Rightarrow \mathop{\mathrm{\mathbf{CAlg}}}_k(\mathcal{O}(G),-)\] induces by Yoneda a morphism \[\delta:\mathcal{O}(G)\to\mathcal{O}(G)\otimes_k\mathcal{O}(G)\] of \(k\)-algebras, called the comultiplication of \(\mathcal{O}(G)\),

  • the group object unit \[\text{unit}:\mathop{\mathrm{\mathbf{CAlg}}}_k(k,-)\Rightarrow\mathop{\mathrm{\mathbf{CAlg}}}_k(\mathcal{O}(G),-)\] induces by Yoneda a morphism \[\epsilon:\mathcal{O}(G)\to k\] of \(k\)-algebras, called the counit of \(\mathcal{O}(G)\),

  • and the group object inverse \[\text{inv}:\mathop{\mathrm{\mathbf{CAlg}}}_k(\mathcal{O}(G),-)\Rightarrow\mathop{\mathrm{\mathbf{CAlg}}}_k(\mathcal{O}(G),-)\] induces by Yoneda a morphism \[i:\mathcal{O}(G)\to \mathcal{O}(G)\] of \(k\)-algebras, called the antipode of \(\mathcal{O}(G)\).

  The group object interaction requirements between \(\text{mult},\text{unit}\) and \(\text{inv}\) translate then precisely to the requirements that the \((\delta,\epsilon,i)\) need to satisfy to endow the commutative \(k\)-algebra \(\mathcal{O}(G)\) with the structure of a cogroup object in the category \(\mathop{\mathrm{\mathbf{CAlg}}}_k\), equivalently a commutative Hopf \(k\)-algebra.

To sum up, we have the equivalent perspectives \[\begin{align} \mathop{\mathrm{\mathbf{AffGrpSch}}}_k\simeq\mathop{\mathrm{\mathbf{AffGrp}}}_k\simeq\mathop{\mathrm{\mathbf{CHopf}}}_k^{\mathop{\mathrm{op}}}. \end{align}\] on the theory. -2.5ex-1ex -.25ex 1.25ex .25ex Classification To present the classification of neutralized Tannakian categories I will mostly use the functorial language of affine groups over \(k\).

I will denote by \[\mathop{\mathrm{\mathbf{Rep}}}_k^{\mathop{\mathrm{f.d}}}(G)\] the category of finite-dimensional representations of an \(G\) over \(k\). It is a neutral Tannakian category over \(k\) admitting the forgetful functor \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/mglasnyv.png}\label{yusxzkcd}\end{figure}\tag{17}\] to the category of finite-dimensional \(k\)-vector spaces as a fibre functor. Again, as in the case of Galois categories, it turns out that all neutralized Tannakian categories are of this form.

As in the theory of Galois categories, we have a reconstruction and a recognition aspect of the theory:

• Reconstruction: For any affine group \(G\), we have \(G\cong\pi_1^{\mathop{\mathrm{Tan}}}(\mathop{\mathrm{\mathbf{Rep}}}_k^{\mathop{\mathrm{f.d}}}(G),U)\).

• Recognition: A neutralized Tannakian category \((\mathbb{T},\omega)\) factorizes through a monoidal equivalence \[\begin{tikzcd}[row sep=large, column sep=large] \mathbb{T}\arrow[r, "\simeq_{\otimes}"]\arrow[d, "\omega"']& \mathop{\mathrm{\mathbf{Rep}}}_k^{\mathop{\mathrm{f.d}}}\left(\pi_1^{\mathop{\mathrm{Tan}}}(\mathbb{T},\omega)\right) \arrow[ld, "U"]\\ \mathop{\mathrm{\mathbf{Vec}}}_k^{\mathop{\mathrm{f.d}}} \end{tikzcd}\]

The main result of the section is the following.

2.0.4 The result↩︎

2.0.4.1 The (\(2\)-) functor \(\gamma:\mathop{\mathrm{\mathbf{NTan}}}_{k,*}\to\mathop{\mathrm{\mathbf{Gal}}}_{*}\)

The most important result in [1], making the connection between Tannakian and Galois categories possible, is the following.

As noted in Remark \(\ref{classep}\), for a field \(k\), Carboni’s construction associates to the pre-Tannakian category \(\mathop{\mathrm{\mathbf{Vec}}}_k^{\mathop{\mathrm{f.d}}}\) the Galois category \[\mathop{\mathrm{CSep}}_{\mathop{\mathrm{comon}}}(\mathop{\mathrm{\mathbf{Vec}}}_k)=\mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}}=\mathop{\mathrm{CSep}}_{\mathop{\mathrm{mon}}}(\mathop{\mathrm{\mathbf{Vec}}}_k)^{\mathop{\mathrm{op}}}\] dual to the category of finite classically separable algebras over \(k\).

It is also proved, in Theorem 2 of Section 6 of [1], that Carboni’s construction associates, furthermore, to any fibre functor \[\omega:\mathbb{T}\to\mathop{\mathrm{\mathbf{Vec}}}_k^{\mathop{\mathrm{f.d}}}\] on a neutral Tannakian category \(\mathbb{T}\) over \(k\), a fibre functor \[\mathop{\mathrm{CSep}}_{\mathop{\mathrm{comon}}}(\omega):\mathop{\mathrm{CSep}}_{\mathop{\mathrm{comon}}}(\mathbb{T})\to \mathop{\mathrm{\mathbf{Set}}}_f\] on the Galois category \(\mathop{\mathrm{CSep}}_{\mathop{\mathrm{comon}}}(\mathbb{T})\). It in fact follows that Carboni’s construction interacts well with the definitions of the (\(2\)-) categories of neutralized Tannakian and Galois categories given earlier in this article, inducing a (\(2\)-) functor \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/bguwvamz.png}\label{zsobupnm}\end{figure}\tag{18}\] from the (\(2\)-) category \(\mathop{\mathrm{\mathbf{NTan}}}_{k,*}\) of neutralized Tannakian categories over \(k\) to the (\(2\)-) category \(\mathop{\mathrm{\mathbf{Gal}}}_{*}\) of neutralized Galois categories.

Next let me remind the reader of the following definitions.

We have identifications \[\begin{align} \mathop{\mathrm{\mathbf{AlgGrp}}}_k\simeq \mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}^{\mathop{\mathrm{op}}})\simeq \mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}^{\mathop{\mathrm{op}}} \end{align}\] and \[\begin{align} \label{quotet} \mathop{\mathrm{\mathbf{\acute{E}tGrp}}}_k\simeq \mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}}). \end{align}\tag{19}\] Next recall (say from Section 6.5 of [10]) that any finitely generated \(k\)-algebra \(A\) admits a maximal separable subalgebra \[\pi_0(A).\]

For any algebraic \(k\)-group \(G\), the inclusion \[\pi_0(\mathcal{O}(G))\to \mathcal{O}(G)\] makes \(\mathcal{O}(G)\) faithfully flat over \(\pi_0(\mathcal{O}(G))\) and thus the corresponding \[q_{G}:G\to G_{\mathop{\mathrm{\acute{e}t}}}:=\mathop{\mathrm{\mathbf{CAlg}}}_k(\pi_0(\mathcal{O}(G)),-)\] is a quotient map.

This construction of the étale quotient is functorial, inducing a functor \[(-)_{\mathop{\mathrm{\acute{e}t}}}:\mathop{\mathrm{\mathbf{AlgGrp}}}_k\to\mathop{\mathrm{\mathbf{\acute{E}tGrp}}}_{k}\] whose dual I will denote by \[\pi_0:\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\to\mathbf{Cogrp}(\mathop{\mathrm{\mathbf{FCSep}}}_k)\cong\mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{FCSep}}}_k^{\mathop{\mathrm{op}}})^{\mathop{\mathrm{op}}}.\]

Now going back to assuming \(k\) to be separably closed, the functor \(\pi_0\) induces a functor \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/abkrsoiu.png}\label{mdjtqbke}\end{figure}\tag{21}\] landing in the dual of the category of finite groups. I am going to be interested in the \(\mathop{\mathrm{\mathbf{Ind}}}\)-completion of this functor. Notice that \[\mathop{\mathrm{\mathbf{Ind}}}\mathop{\mathrm{\mathbf{Grp}}}_f^{\mathop{\mathrm{op}}}\simeq \mathop{\mathrm{\mathbf{Pro}}}(\mathop{\mathrm{\mathbf{Grp}}}_f)^{\mathop{\mathrm{op}}}\simeq\mathop{\mathrm{\mathbf{StoneGrp}}}^{\mathop{\mathrm{op}}}.\] The \(\mathop{\mathrm{\mathbf{Ind}}}\)-completion of \(\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\) is the full category of commutative Hopf algebras over \(k\); this is certainly well-known but as I have not found a reference for this in the literature let me sketch the proof. So firstly notice that \(\mathop{\mathrm{\mathbf{CAlg}}}_k\) is a locally finitely presentable category, whose essentially small full subcategory consisting of finitely presentable objects identifies with the category \(\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}\) of finitely generated commutative \(k\)-algebras; it is then a consequence of Gabriel-Ulmer duality that \(\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}\) has finite colimits and that the embedding \[\begin{tikzcd}[row sep=large, column sep=large] \mathop{\mathrm{\mathbf{CAlg}}}_k\\ \mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}\arrow[u, hook, "\iota"] \end{tikzcd}\] preserves them, as it realizes \(\mathop{\mathrm{\mathbf{CAlg}}}_k\) as \(\mathop{\mathrm{\mathbf{CAlg}}}_k\simeq \mathop{\mathrm{\mathbf{Ind}}}\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}\), the free cocompletion of \(\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}\) under filtered colimits. Hence the fully faithful \[\iota^{\mathop{\mathrm{op}}}:\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}^{\mathop{\mathrm{op}}}\to \mathop{\mathrm{\mathbf{CAlg}}}_k^{\mathop{\mathrm{op}}}\] preserves finite limits and therefore group objects can be defined to give a functor \[\begin{align} \mathop{\mathrm{\mathbf{Grp}}}(\iota^{\mathop{\mathrm{op}}}):\mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}^{\mathop{\mathrm{op}}})&\to& \mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{CAlg}}}_k^{\mathop{\mathrm{op}}})\\ G:\mathbb{L}_{\mathop{\mathrm{\mathbf{Grp}}}}\to \mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}^{\mathop{\mathrm{op}}}&\mapsto& \iota^{\mathop{\mathrm{op}}}\circ G, \end{align}\] where \(\mathbb{L}_{\mathop{\mathrm{\mathbf{Grp}}}}\) denotes the Lawvere theory of groups. Next, notice that the functor \(\mathop{\mathrm{\mathbf{Grp}}}(\iota^{\mathop{\mathrm{op}}})\) is fully faithful, as we clearly have \[\begin{align} \mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{CAlg}}}_k^{\mathop{\mathrm{op}}})\left(\iota^{\mathop{\mathrm{op}}}\circ G, \iota^{\mathop{\mathrm{op}}}\circ H\right)&\cong&\int_{\ell\in \mathbb{L}_{\mathop{\mathrm{\mathbf{Grp}}}}}\mathop{\mathrm{\mathbf{CAlg}}}_k^{\mathop{\mathrm{op}}}\left(\iota^{\mathop{\mathrm{op}}}(G(\ell^{\dagger})), \iota^{\mathop{\mathrm{op}}}(H(\ell^{\ddagger}))\right)\\ &\cong& \int_{\ell\in\mathbb{L}_{\mathop{\mathrm{\mathbf{Grp}}}}}\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}^{\mathop{\mathrm{op}}}(G(\ell^{\dagger}),H(\ell^{\ddagger}))\\ &\cong& \mathop{\mathrm{\mathbf{Grp}}}(\mathop{\mathrm{\mathbf{CAlg}}}_{k,\mathop{\mathrm{f.g}}}^{\mathop{\mathrm{op}}})(G,H), \end{align}\] where the second isomorphism follows from the fact that \(\iota^{\mathop{\mathrm{op}}}\) is fully faithful. The dual of this functor is the fully faithful functor \[\begin{tikzcd}[row sep=large, column sep=large] \mathop{\mathrm{\mathbf{CHopf}}}_k\\ \mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\arrow[u, hook, "\mathop{\mathrm{\mathbf{Grp}}}(\iota^{\mathop{\mathrm{op}}})^{\mathop{\mathrm{op}}}"] \end{tikzcd}\]

Thus the \(\mathop{\mathrm{\mathbf{Ind}}}\)-completion of the functor \(\pi_0\) is a functor \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/iywngkqm.png}\label{kszpwohy}\end{figure}\tag{22}\]

Now given a neutralized Tannakian category \((\mathbb{T},\omega)\), and writing the coendomorphism commutative Hopf algebra \[\mathcal{O}\left(\pi_1^{\mathop{\mathrm{Tan}}}(\mathbb{T},\omega)\right)\cong\varinjlim_i \mathcal{O}\left(\pi^{\mathop{\mathrm{Tan}}}_1(\mathbb{T},\omega)\right)_i\] as the filtered colimit of its finitely generated Hopf subalgebras \(\mathcal{O}\left(\pi^{\mathop{\mathrm{Tan}}}_1(\mathbb{T},\omega)\right)_i\), we can conclude, using Carboni’s theory, that the fundamental group of the Galois fibre functor constructed from \((\mathbb{T},\omega)\) using commutative separable monoids, can be realized \[\begin{align} \pi_1^{\mathop{\mathrm{Gal}}}\left(\mathop{\mathrm{CSep}}(\mathbb{T})_{\mathop{\mathrm{comon}}},\mathop{\mathrm{CSep}}(\omega)_{\mathop{\mathrm{comon}}}\right)\cong \varprojlim_{i}\mathop{\mathrm{\mathbf{CAlg}}}_k\left(\pi_0\left(\mathcal{O}(\pi_1^{\mathop{\mathrm{Tan}}}(\mathbb{T},\omega)_i,k\right) \right) \end{align}\] as the cofiltered limit of the finite groups arising as the \(k\)-points of the maximal separable subalgebras of each \(\mathcal{O}\left(\pi^{\mathop{\mathrm{Tan}}}_1(\mathbb{T},\omega)\right)_i\), inducing an identification \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/exztwbhy.png}\label{briheajk}\end{figure}\tag{23}\] of \(\gamma\) with the composite \(\mathop{\mathrm{\mathfrak{CONT}}}\circ\mathop{\mathrm{\mathbf{Ind}}}\pi_0(-)_k\circ\mathop{\mathrm{\mathfrak{AUT}}}^{\vee}\).

2.0.4.2 The fully faithful left (\(2\)-) adjoint to \(\gamma\)

Let \(G\) be a finite group and let \(k\) be a general field. Recall that the constant algebraic group \(G_k\) has as underlying scheme the disjoint union of copies of \(\mathop{\mathrm{Spec}}(k)\) indexed by the elements of \(G\), i.e. \[|G_k|=\bigsqcup_{g\in G}\mathop{\mathrm{Spec}}(k);\] its coordinate algebra \(\mathcal{O}(G_k)\) is just the set of functions \(\mathop{\mathrm{\mathbf{Set}}}(G,k)\) with is usual algebra structure and thus as an affine group over \(k\), \(G_k\) is the functor \[\begin{align} G_k:\mathop{\mathrm{\mathbf{CAlg}}}_k&\to&\mathop{\mathrm{\mathbf{Grp}}}\\ A&\mapsto& \mathop{\mathrm{\mathbf{CAlg}}}_k(\mathop{\mathrm{\mathbf{Set}}}(G,k),A). \end{align}\]

For the rest of the section assume \(k\) to be separably closed. The assignment \(G\mapsto G_k\) induces a functor \[(-)_k:\mathop{\mathrm{\mathbf{Grp}}}_f\to\mathop{\mathrm{\mathbf{AlgGrp}}}_k\] which is fully faithful, corresponding to the full embedding of étale groups in algebraic groups. I will denote its dual functor by \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ewackflm.png}\label{dthyqgpk}\end{figure}\tag{24}\] It is easy to see that \(k^{(-)}\) is left adjoint to \(\pi_0(-)_k\), i.e. we have a coreflection \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\arrow[d,shift left=1.4ex, "\pi_0(-)_k"]\\ \mathop{\mathrm{\mathbf{Grp}}}_f^{\mathop{\mathrm{op}}}\arrow[u,hook,shift left=1.4ex, "k^{( -)}", "\dashv"'] \end{tikzcd}\] of \(\mathop{\mathrm{\mathbf{Grp}}}_f^{\mathop{\mathrm{op}}}\) in \(\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\); indeed, we have, naturally in \(\Gamma\in \mathop{\mathrm{\mathbf{Grp}}}\) and \(H\in\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\), the isomorphisms \[\begin{align} \mathop{\mathrm{\mathbf{Grp}}}_f^{\mathop{\mathrm{op}}}\left(\Gamma,\pi_0(H)_k\right)&\cong& \mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\left(k^{\Gamma},k^{\pi_0(H)_k}\right)\\ &\cong&\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\left(k^{\Gamma},\pi_0(H)\right)\\ &\cong&\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\left(k^{\Gamma},H\right), \end{align}\] with the second isomorphism following from the fact that for any algebraic \(k\)-group \(G\), \(G_{\mathop{\mathrm{\acute{e}t}}}\) is constant as we work with \(k\) separably closed, and the last isomorphism following from the fact that \(k^{G}\) is separable and algebra homomorphisms from a separable algebra to \(H\) factor uniquely through the inclusion \(\pi_0(H)\subseteq H\).

Now since both \(\mathop{\mathrm{\mathbf{CHopf}}}_{k,\mathop{\mathrm{f.g}}}\) and \(\mathop{\mathrm{\mathbf{Grp}}}_f^{\mathop{\mathrm{op}}}\) have finite colimits, the coreflection \(\mathop{\mathrm{\mathbf{Ind}}}\)-completes to produce a coreflection \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{CHopf}}}_{k}\arrow[d,shift left=1.4ex, "\mathop{\mathrm{\mathbf{Ind}}}\pi_0(-)_k"]\\ \mathop{\mathrm{\mathbf{StoneGrp}}}^{\mathop{\mathrm{op}}}\arrow[u,hook,shift left=1.4ex, "\mathop{\mathrm{\mathbf{Ind}}}k^{( -)}", "\dashv"'] \end{tikzcd}\]

We can thus define the fully faithful (\(2\)-) adjoint of \(\gamma\) \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/dkpsyezg.png}\label{hnigzqju}\end{figure}\tag{25}\] as the composite \((-)_{\mathop{\mathrm{lin}}}:=\mathop{\mathrm{\mathfrak{COMOD}}}\circ\mathop{\mathrm{\mathbf{Ind}}}k^{(-)}\circ\mathop{\mathrm{\mathfrak{AUT}}}\), inducing a \(2\)-coreflection \[\begin{tikzcd}[row sep=large] \mathop{\mathrm{\mathbf{NTan}}}_{k,*}\arrow[d,shift left=1ex, "\gamma"]\\ \mathop{\mathrm{\mathbf{Gal}}}_{*}\arrow[u,hook,shift left=1.4ex, "\dashv"', "(-)_{\mathop{\mathrm{lin}}}"] \end{tikzcd}\]

References↩︎

---

1. math.georgios@gmail.com↩︎

2. Notice that the \(2\)-equivalence between Tannakian \(k\)-categories and affine fpqc gerbes over \(\mathop{\mathrm{Spec}}k\) is contravariant, so it is co- reflective behaviour that should be expected on this side of the theory.↩︎

3. Notice however, that any two fibre functors on a Galois category \(\mathcal{C}\) are (non-canonically) isomorphic.↩︎

4. I will not go into the definitions of these morphisms in detail as that would divert me from the main goal of this paper, especially because in the case of interest for us, namely when \(X=\mathop{\mathrm{Spec}}k\), finite étale \(X\)-schemes take a much simpler form - see the next remark.↩︎

5. Recall that a commutative \(k\)-algebra \(A\) is finite over \(k\) if it is finite-dimensional as a \(k\)-vector space.↩︎

6. These structures are called hypergraph* categories in modern literature.*↩︎