Freyd cover

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Discrete and concrete objects
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[[!include discrete and concrete objects - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The **Freyd cover** $\overline{\mathcal{T}}$ of a [[topos]] $\mathcal{T}$, or more generally of a [[category]] with a [[terminal object]],  is a  way to turn $\mathcal{T}$, viewed as an 'abstract' category, into a 'concrete' category of structured sets by sewing it together with [[Set]] along the [[global section functor]].

As $\overline{\mathcal{T}}$ comes equipped with two well-behaved projection functors the resulting category has many good logical properties making the construction an important tool in [[type theory]] and theoretical computer science.

The Freyd cover is sometimes known as the **Sierpinski cone** or **scone**, because in [[topos theory]] it behaves similarly to the [[cone]] on a space, but with the [[interval]] $[0,1]$ replaced by the [[Sierpinski space]].

## Definition

Let $\mathcal{T}$ be a category with a terminal object and $\Gamma = \mathcal{T} (1, -)$ the global section functor . The **Freyd cover** of $\mathcal{T}$ is the category  $\overline{\mathcal{T}}$ whose objects are triples $(X, \xi, U)$ where:

* $X$ is a set
* $U$ is an object of $\mathcal{T}$
* $\xi$ is a function $X \to \Gamma (U)$

A morphism from $(X,\xi, U)$ to $(Y,\eta , V)$ is a pair of morphisms $(\varphi :X\to Y, t:U\to V)$ with $\varphi\in Set$ and $t\in \mathcal{T}$ such that $\eta\varphi=\Gamma(t)\xi$.

### Remark

The construction is a special case of [[Artin gluing]]: i.e.  $\overline{\mathcal{T}}$ is the [[comma category]] $Set \downarrow \Gamma$ with $\Gamma = \mathcal{T} (1, -)$.

## Properties

### Relation to the initial topos

One of the first applications of the Freyd cover was to deduce facts about the initial topos (initial with respect to [[logical morphism|logical morphisms]] --- also called the [[free topos]]). They were originally proved by syntactic means; the conceptual proofs of the lemma and theorem below are due to Freyd.

+-- {: .num_lemma}
###### Lemma
For any category $C$ with a [[terminal object]] $\mathbf{1}$, the terminal object of the Freyd cover $\widehat{C}$ is [[small-projective]], i.e., the representable $\Gamma = \widehat{C}(1, -) \colon \widehat{C} \to Set$ preserves any colimits that exist.
=--

+-- {: .proof}
###### Proof
To check that $\Gamma^{op} \colon \widehat{C}^{op} \to Set^{op}$ preserves limits, it suffices to check that the composite

\begin{center}
\begin{tikzcd}
\widehat{C}^{op} \arrow[r, "\Gamma^{op}"] & \mathrm{Set}^{op} \arrow[r, "2^-"] & \mathrm{Set}
\end{tikzcd}
\end{center}

preserves limits, because the contravariant power set functor $P = 2^-$ is monadic. But it is easily checked that this composite is the contravariant representable given by $(2, \mathbf{1}, 2 \to \Gamma(\mathbf{1}))$.
=--

+-- {: .num_theorem}
###### Theorem
The terminal object in the initial topos $\mathcal{T}$ is [[connected object|connected]] and [[projective object|projective]] in the sense that $\Gamma = \hom(1, -) \colon \mathcal{T} \to Set$ preserves finite colimits.
=--

+-- {: .proof}
###### Proof
We divide the argument into three segments:

* The hom-functor preserves finite limits, so by general properties of Artin gluing, the Freyd cover $\widehat{\mathcal{T}}$ is also a topos. Observe that $\mathcal{T}$ is equivalent to the slice $\widehat{\mathcal{T}}/M$ where $M$ is the object $(\emptyset, \mathbf{1}, \emptyset \to \Gamma(\mathbf{1}))$. Since pulling back to a slice is a logical functor, we have a logical functor
$$\pi \colon \widehat{\mathcal{T}} \to \mathcal{T}$$
Since $\mathcal{T}$ is initial, $\pi$ is a retraction for the unique logical functor $i \colon \mathcal{T} \to \widehat{\mathcal{T}}$.

* We have maps $\mathcal{T}(1, -) \to \widehat{\mathcal{T}}(i 1, i-) \cong \widehat{\mathcal{T}}(1, i-)$ (the isomorphism comes from $i 1 \cong 1$, which is clear since $i$ is logical), and $\widehat{\mathcal{T}}(1, i-) \to \mathcal{T}(\pi 1, \pi i-) \cong \mathcal{T}(1, -)$ since $\pi$ is logical and retracts $i$. Their composite must be the identity on $\mathcal{T}(1, -)$, because there is only one such endomorphism, using the Yoneda lemma and terminality of $1$.

Finally, since $\mathcal{T}(1, -)$ is a retract of a functor $\widehat{\mathcal{T}}(1, i-)$ that preserves finite colimits (by the lemma, and the fact that the logical functor $i$ preserves finite colimits), it must also preserve finite colimits.
=--

This is important because it implies that the [[internal logic]] of the free topos (which is exactly "intuitionistic higher-order logic") satisfies the following properties:

* The *disjunction property*: if "P [[or]] Q" is provable in the empty [[context]], then either P is so provable, or Q is so provable.  (Note that this clearly fails in the presence of [[excluded middle]].)

* The *existence property*: if "there exists an $x\in A$ such that $P(x)$" is provable in the empty context, then there exists a [[global element]] $x\colon 1\to A$ such that $P(x)$ is provable in the empty context.  (Again, this is clearly a [[constructive mathematics|constructivity]] property.)

* The *negation property*: False is not provable in the empty context.

* All numerals in the free topos are "standard", i.e., the global sections functor $\Gamma = \hom(1, -): \mathcal{T} \to Set$ preserves the [[natural numbers object]] $N$ (because $N$ can be characterized in terms of finite colimits and $1$, by a theorem of Freyd).

### As a local topos
 {#AsALocalTopos}

The Freyd cover of a [[topos]] is a [[local topos]], and in fact freely so. Every local topos is a [[retract]] of a Freyd cover.

See ([Johnstone, lemma C3.6.4](#Johnstone)).

### As a fibration

The Freyd cover $\overline{\mathcal{T}}$ is fibered over $\mathcal{T}$ as it arises equivalently by change of base of the [[codomain fibration]] of [[Set]] along the [[global section functor]] $\Gamma$.

See ([Jacobs, p.57](#Jacobs)).

## Related entries

* [[Artin gluing]]

* [[comma category]]

* [[Sierpinski topos]]

## References

Some of the above material is taken from

* [[Tom Leinster]], [reply at MathOverflow](http://mathoverflow.net/questions/12136/freyd-cover-of-a-category)

The following two n-café blog posts provide an introductory discussion of the scone construction from a geometric and a logical perspective:

* [[Mike Shulman]], _Discreteness, Concreteness, Fibrations, and Scones_ , November 2011. ([link](https://golem.ph.utexas.edu/category/2011/11/discreteness_concreteness_fibr.html))

* [[Mike Shulman]], _Scones, Logical Relations, and Parametricity_ , April 2013. ([link](https://golem.ph.utexas.edu/category/2013/04/scones_logical_relations_and_p.html))

You can find more on [[Artin gluing]] in this important (and nice) paper:

* [[Aurelio Carboni]], [[Peter Johnstone]], _Connected limits, familial representability and Artin glueing_ , Mathematical Structures in Computer Science **5** (1995) pp.441-459.

plus

* [[Aurelio Carboni]], [[Peter Johnstone]], _Corrigenda to 'Connected limits...'_ , Mathematical Structures in Computer Science **14** (2004) pp.185-187.

See also section C3.6 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. 2_ . Oxford UP 2002.
  {#Johnstone}

* [[Bart Jacobs]], _Categorical Logic and Type Theory_ , Elsevier Amsterdam 1999. {#Jacobs}

The following paper studies logical properties of the Freyd cover:

* [[Ieke Moerdijk]], _On the Freyd Cover of a Topos_ , Notre Dame J. Formal Logic **24** no.4 (1983) pp.517-526. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093870454))

The argument given above for the properties of the free topos is an amplification of

* [[Peter Freyd|P. Freyd]], A. Scedrov, _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990.  (1.(10)31, p.192)

For more on the free topos and the first appearance in print of Freyd's observation see

* [[Joachim Lambek|J. Lambek]], P. J. Scott, _Intuitionist Type Theory and the Free Topos_ , JPAA **19** (1980) pp.215-257.

[[!redirects scone]]
[[!redirects Sierpinski cone]]


Freyd multicategory

# Freyd multicategories

* toc
{: toc}

## Idea

A **Freyd multicategory** is to a [[multicategory]] as a [[Freyd category]] is to a [[monoidal category]].

## Definition

A **Freyd multicategory** consists of

* A [[multicategory]] $\mathcal{V}$ (usually [[cartesian multicategory|cartesian]])
* A [[premulticategory]] $\mathcal{C}$
* A functor $return : \mathcal{V}\to\mathcal{C}$ that is the [[identity-on-objects functor|identity on objects]] and preserves central morphisms.

## Relation to Freyd categories

A Freyd multicategory that has all tensor products and units, in a usual multicategorical sense that are preserved by $return$, is equivalent to a [[Freyd category]].

## References

* [[Sam Staton]] and [[Paul Blain Levy]], *Universal properties of impure programing languages*.  POPL '13: Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, 2013, [doi](https://doi.org/10.1145/2429069.2429091)

[[!redirects Freyd multicategories]]
[[!redirects Freyd-multicategory]]
[[!redirects Freyd-multicategories]]


Freyd's adjoint functor theorem

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Freyd's adjoint functor theorem (FAFT) is a fine thing when working classically but at the same time a source of inconstructivity. It ensures that under certain assumptions on a functor F : AA->BB all presheaves of the form BB(F(-),B) or all covariant presheaves of the form BB(B,F(-)) are representable. But for getting a right or left adjoint to F one has to CHOOSE for every B a representing object R(B) or L(B), respectively, i.e. BB(F(-),B) \cong BB(-,R(B)) or BB(B,F(-)) \cong BB(L(B),_), resp. Since BB typically will be large this requires GLOBAL CHOICE, i.e. choice for classes. This principle is independent even from ZFC.

For example using FAFT G.Plotkin has shown that there exist free commutative idempotent monoids in Domains, the so called Plotkin powerdomains. But to explicitate representatives even for a restricted class of domains (SFP objects) has required quite some additional effort and ingenuity.

In geometric topos theory it goes withut saying that a (finite limit preserving) colimit preserving functor between Grothendieck toposes has always a right adjoint. The point is that the solution set condition follows from the assumption that the involved toposes are Grothendieck and thus have small generating families. But for getting the right adjoints one has to use global choice.

When working relative to a base topos SS one has to employ a fibred version of FAFT for fibrations over SS. The fibred right adjoints will not be split even if the original functors was split. That is one of the reasons why one works with fibrations and cartesian functors instead of split fibrations and split cartesian functors. For proving fibred FAFT one hast to employ strong choice principles on the meta-level, of course.

It might be interesting to investigate how choice can be avoided in these cases using the work of Ahrens, Kapulkin and Shulman on "Univalent Categories and Rezk completion".

Thomas Streicher
=--


Freyd-Mitchell embedding theorem

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Additive and abelian categories
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[[!include additive and abelian categories - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Freyd--Mitchell embedding theorem_ says that every [[abelian category]] $\mathcal{A}$ is a [[full subcategory]] of a [[category of modules]] over some [[ring]] $R$, such that the [[fully faithful functor|embedding functor]] $\mathcal{A} \hookrightarrow R Mod$ is an [[exact functor]].

## Details

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###### Remark

It is easy to see that not every abelian category is _[[equivalence of categories|equivalent]]_ to $R$[[Mod]] for some [[ring]] $R$.  The reason is that $R Mod$ has all [[small category|small]] [[limits]] and [[colimits]].  But for instance, for $R$ [[noetherian ring|Noetherian]], the category of [[finitely generated module|finitely generated]] $R$-modules is an abelian category but lacks these properties.

=--

However, we have:

\begin{theorem}
\label{MitchellEmbeddingTheorem}
**(Mitchell embedding theorem)**
\linebreak
Every [[small category|small]] [[abelian category]] admits a [[full functor|full]], [[faithful functor|faithful]] and [[exact functor|exact]] functor to the category $R$[[Mod]] for some [[ring]] $R$.
\end{theorem}

This result can be found as Theorem 7.34 on page 150 of ([Freyd](#Freyd)).  (The terminology there is a bit outdated, in that it calls an abelian category "fully abelian" if it admits a full and faithful exact functor to a category of $R$-modules.)  A pedagogical discussion is in section 1.6 of ([Weibel](#Weibel)). See also ([Wikipedia](http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem)) for the idea of the proof.

+-- {: .proof #Proof}
###### Proof

(...)

=--

We can also characterize which abelian categories _are_ equivalent to a category of $R$-modules:

+-- {: .num_theorem}
###### Theorem

Let $C$ be an [[abelian category]].  If $C$ has all [[small category|small]] [[coproducts]] and has a [[compact object|compact]] [[projective object|projective]] [[generator]], then $C \simeq R Mod$ for some ring $R$.

In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any compact projective generator.   Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.

=--

This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of ([Freyd](#Freyd)).
The first part of this theorem can also be found as Prop. 2.1.7 in ([Ginzburg](#Ginzburg)).  Conversely, it is easy to see that $R$ is a compact projective generator of $R Mod$.

Going further, we can try to characterize functors between categories of $R$-modules that come from tensoring with bimodules.  Here we have

+-- {: .num_theorem}
###### Watts' Theorem

If $B$ is an an $S$-$R$-bimodule, the [[tensor product]] functor

$$
  B \otimes_R -\colon R Mod \to S Mod
$$

is [[right exact functor|right exact]] and preserves [[small category|small]] [[coproducts]].  Conversely, if $F\colon Mod_R \to Mod_S$ is right exact and that preserves small coproducts, it is naturally isomorphic to $B \otimes_R -$ where $B$ is the $S$-$R$-bimodule $F R$.

=--

This theorem was more or less simultaneously proved by Watts and Eilenberg; a generalization is proved in ([Nyman-Smith](#NymanSmith)), and references to the original papers can be found there.

Going still further we should be able to obtain a nice theorem describing the [[image]] of the embedding of the [[2-category]] of

* rings
* bimodules
* bimodule homomorphisms

into the strict 2-category of

* abelian categories
* right exact functors
* natural transformations.

For more discussion see the [$n$-Cafe](http://golem.ph.utexas.edu/category/2007/08/questions_about_modules.html).

## Related concepts

* [stable model category -- as A-infinity algebroid module categories](stable+model+category#AsCategoriesOfModules)

* [[reconstruction theorem]]

* [[derived category]]

## References

Original articles:

* [[Barry Mitchell]], *The Full Imbedding Theorem*, American Journal of Mathematics **86** 3 (1964) 619-637 [[doi:10.2307/2373027](https://doi.org/10.2307/2373027)]

Textbook account:

* {#Freyd} [[Peter Freyd]], *Abelian Categories*, Harper and Row (1964), Reprints in Theory and Applications of Categories **3** (2003) 23-164 [[tac:tr3](http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html)]

Details on the proof and its variants are also in

* {#Weibel} [[Charles Weibel]], section 1.6 of: _[[An Introduction to Homological Algebra]]_

and

* {#Ginzburg} [[Victor Ginzburg]], _Lectures on noncommutative geometry_ ([pdf](http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf))

* {#NymanSmith} A. Nyman , S. Paul Smith, _A generalization of Watts's Theorem: Right exact functors on module categories_ ([arXiv:0806.0832](http://arxiv.org/abs/0806.0832))

An introductory survey is for instance also in section 3 of

* Geillan Aly, _Abelian Categories and the Freyd-Mitchell Embedding Theorem_ ([pdf](http://www.u.arizona.edu/~geillan/research/ab_categories.pdf))

See also

* Wikipedia, _[Mitchell's embedding theorem](http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem)_

[[!redirects Freyd-Mitchell embedding theorem]]
[[!redirects Freyd–Mitchell embedding theorem]]
[[!redirects Freyd--Mitchell embedding theorem]]
[[!redirects Freyd-Mitchell embedding]]
[[!redirects Freyd–Mitchell embedding]]
[[!redirects Freyd--Mitchell embedding]]
[[!redirects Freyd-Mitchell theorem]]
[[!redirects Freyd–Mitchell theorem]]
[[!redirects Freyd--Mitchell theorem]]

[[!redirects Mitchell theorem]]
[[!redirects Mitchell's theorem]]
[[!redirects Mitchell's theorem]]

[[!redirects Watt's theorem]]
[[!redirects Watt theorem]]

[[!redirects Mitchell embedding theorem]]


friction

## Idea

(some discussion should eventually go here, for the moment this entry just serves as a container for the (maybe) interesting piece of ancient history of thought recorded below)

## History

Aristotle may have been either ignorant or dismissive of friction but in the Epicurean school exemplified in the writings by Lucretious it is not, he says that atoms 'are in perpetual motion at enormous speed, since in the void they get no resistance from the medium, and when they collide they can only be deflected, not halted'.

This is in the lines 80-332 in Book 2, section 4 of

* Lucretius, _De Rerum Natura_ ([web](http://classics.mit.edu/Carus/nature_things.2.ii.html))

The author there argues that atoms are in constant motion (all in the English translation by William Leonard):

> For far beneath the ken of senses lies

> The nature of those ultimates of the world;

> And so, since those themselves thou canst not see,

> Their motion also must they veil from men

and that they are constantly colliding with another:

> Inveterately plied by motions mixed,

> Some, at their jamming, bound aback and leave

> Huge gaps between, and some from off the blow

> Are hurried about with spaces small between.

But they're able to be stay together to form compounds, but they're still in motion:

> And all which, brought together with slight gaps,

> In more condensed union bound aback,

> Linked by their own all intertangled shapes,-

> These form the irrefragable roots of rocks

> And the brute bulks of iron

and that the motion must be unpredictable, (Empson uses the word swerve here), for otherwise they'd never collide and nothing would be then created.

> For were it not their wont

> Thuswise to swerve,

> down would they fall, each one,

> Like drops of rain, through the unbottomed void;

> And then collisions ne'er could be nor blows

> Among the primal elements; and thus

> Nature would never have created aught


Fridrich Valach

* [personal page](https://sites.google.com/view/fvalach)

## Selected writings

On [[non-abelian T-duality]] in [[type II supergravity]]:

* [[Pavol Ševera]], [[Fridrich Valach]], _Courant algebroids, Poisson-Lie T-duality, and type II supergravities_ ([arXiv:1810.07763](https://arxiv.org/abs/1810.07763))

On [[duality in physics|duality]] for [[higher gauge theories]], subsuming [[non-abelian T-duality]] and [[electric-magnetic duality]], via [[boundary conditions]] for [[AKSZ sigma-models]]:

* [[Ján Pulmann]], [[Pavol Ševera]], [[Fridrich Valach]], _A non-abelian duality for (higher) gauge theories_ ([arXiv:1909.06151](https://arxiv.org/abs/1909.06151))

A [[Lagrangian density]] for the [[boson|bosnic sector]] of [[type II supergravity]] making the nature of the [[pregeometric RR-fields]] and their [[self-dual higher gauge theory|self-duality]] manifest:

* [[Karapet Mkrtchyan]], [[Fridrich Valach]], *Democratic actions for type II supergravities*, Phys.Rev.D **107** 6 (2023) 066027 [[arXiv:2207.00626](https://arxiv.org/abs/2207.00626), [doi:10.1103/PhysRevD.107.066027](https://doi.org/10.1103/PhysRevD.107.066027)]

category: people


Friedemann Brandt

* [arXiv page](https://arxiv.org/a/brandt_f_1.html)

## Selected writings

On [[local BRST cohomology]] and [[BV-formalism]]:

* {#BarnichBrandtHenneaux94} [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in the antifield formalism: I. General theorems_, Commun. Math. Phys. **174** (1995) 57-92 [[arXiv:hep-th/9405109](https://arxiv.org/abs/hep-th/9405109), [doi:10.1007/BF02099464](https://doi.org/10.1007/BF02099464)]

* {#BarnichBrandtHenneaux00} [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in gauge theories_, Phys. Rept. **338** (2000) 439-569 [doi:10.1016/S0370-1573(00)00049-1[arXiv:hep-th/0002245](https://arxiv.org/abs/hep-th/0002245)]

and application to [[gravity]] and/or [[Yang-Mills theory]] ([[Einstein-Yang-Mills theory]]):

* [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in Einstein-Yang-Mills theory_, Nucl. Phys. B **455** (1995) 357-408 [doi:10.1016/0550-3213(95)00471-4[arXiv:hep-th/9505173](https://arxiv.org/abs/hep-th/9505173)]

## Related entries

* [[brane scan]]

* [[super Poincare Lie algebra]]

* [[local BRST cohomology]]

category: people


Friederich Mautner

Austrian mathematician (1921 – 2001) who worked in the UK, Eire and the US, largely on [[representation theory]].

* [Wikipedia page](https://en.wikipedia.org/wiki/Friederich_Ignaz_Mautner)

## Related webpages

* [[logicality and invariance]]


Friedhelm Waldhausen

* [Wikipedia entry](http://en.wikipedia.org/wiki/Friedhelm_Waldhausen)

## Selected writings

Introducing [[A-theory]] ([[algebraic K-theory]] of [[suspension spectra]] of [[loop spaces]]):

* [[Friedhelm Waldhausen]], _Algebraic K-theory of spaces_, In: A. Ranicki N.,  Levitt, F. Quinn (eds.), Algebraic and Geometric Topology, Lecture Notes in Mathematics, vol 1126. Springer, Berlin, Heidelberg (1985) ([doi:10.1007/BFb0074449](https://doi.org/10.1007/BFb0074449))

## Related $n$Lab entries

* [[algebraic topology]]

* [[Waldhausen category]]

* [[Waldhausen S-construction]]

* [[Waldhausen K-theory]]

* [[3-manifold]]

category: people

[[!redirects F. Waldhausen]]


Friedlander-Milnor isomorphism conjecture

> under construction

#Contents#
* table of contents
{:toc}

## Statement

Let

* $k$ be an [[algebraically closed field]];

* $G$ be a [[reductive algebraic group]] over $k$;

* $\ell$ a [[prime number]] invertible in $k$ ($\ell \neq char(k)$);

then the canonical morphism

$$
  H^\bullet_{et}(\mathbf{B}G, \mathbb{Z}/\ell\mathbb{Z})
  \longrightarrow
  H^\bullet_{et}(\flat \mathbf{B}G, \mathbb{Z}/\ell\mathbb{Z})
  =
  H^\bullet(B G(k), \mathbb{Z}/\ell\mathbb{Z})
$$

(from the [[etale cohomology]] of the [[moduli stack]] of $G$ (the [[quotient stack]] $\mathbf{B}G \simeq \ast//G$) to the cohomology of its underlying [[discrete group]] of $k$-points) is an [[isomorphism]].

## References
Review includes

* [[Eric Friedlander]], _The Friedlander-Milnor conjecture_ ([pdf](http://retro.seals.ch/cntmng?pid=ensmat-001:2008:54::134))

* [MO discussion](http://mathoverflow.net/q/163250/381)

* [[Fabien Morel]], _On the homology of Lie groups made discrete_, talk, 2012 ([video](http://www.ima.umn.edu/videos/?id=1794))

Original articles include

* [[Eric Friedlander]], Guido Mislin, _Cohomology of classifying spaces of complex Lie groups and related discrete groups_ ([pdf](http://retro.seals.ch/cntmng?pid=comahe-003:1984:59::26))


Friedrich Hirzebruch

* [Wikipedia entry](http://en.wikipedia.org/wiki/Friedrich_Hirzebruch)

## Selected writings

Early discussion of [[characteristic classes]], [[cobordism theory]], [[genera]] and introducing the notion of multiplicative sequences and characteristic series of [[genera]]:

* {#Hirzebruch56} [[Friedrich Hirzebruch]], *Neue topologische Methoden in der algebraischen Geometrie*, Ergebnisse der Mathematik und ihrer Grenzgebiete **9**, Springer (1956) [[doi:10.1007/978-3-662-41083-7](https://doi.org/10.1007/978-3-662-41083-7)]

* {#Hirzebruch66} [[Friedrich Hirzebruch]], *Topological methods in algebraic geometry*, Grundlehren **131**, Springer (1966) [[doi:10.1007/978-3-642-62018-8](https://doi.org/10.1007/978-3-642-62018-8), MR0202713 (34 #2573) Zbl 0843.14009]

Introducing [[topological K-theory]]:

* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], _Riemann-Roch theorems for differentiable manifolds_, Bull.  Amer. Math Soc. vol. 65 (1959) pp. 276-281 ([euclid:bams/1183523205](https://projecteuclid.org/euclid.bams/1183523205))

* {#AtiyahHirzebruch61} [[M. F. Atiyah]], [[F. Hirzebruch]], _Vector bundles and homogeneous spaces_, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7–38 American Mathematical Society, Providence, R.I. (doi:10.1142/9789814401319_0008, [web](http://hirzebruch.mpim-bonn.mpg.de/87/), [MR 0139181](http://www.ams.org/mathscinet-getitem?mr=0139181))

On [[elliptic genera]] and the [[rigidity theorem for elliptic genera]]:

* [[Friedrich Hirzebruch]], _Elliptic Genera of Level $N$ for Complex Manifolds_, In: Bleuler K., Werner M. (eds) _Differential Geometrical Methods in Theoretical Physics_ NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 250. Springer (1988) ([doi:10.1007/978-94-015-7809-7_3](https://doi.org/10.1007/978-94-015-7809-7_3))

On [[genera]], [[elliptic genera]] and [[modular forms]]:

* {#HirzebruchGergerJung92} [[Friedrich Hirzebruch]], Thomas Berger, Rainer Jung, *Manifolds and Modular Forms*, Aspects of Mathematics **20**, Viehweg (1992), Springer (1994) [[doi:10.1007/978-3-663-10726-2](https://doi.org/10.1007/978-3-663-10726-2), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf)]

category: people

[[!redirects F. Hirzebruch]]
[[!redirects Hirzebruch]]


Friedrich Karl Schmidt

* [Wikipedia entry](https://en.wikipedia.org/wiki/Friedrich_Karl_Schmidt)

* Wikipedia entry (deutsch)

## Selected writings

Introducing the [[Hasse-Schmidt derivative]]:

* [[Helmut Hasse]]: *Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von [[Friedrich Karl Schmidt|F. K. Schmidt]] in Jena.)*, Reine Angew. Math.  **177** (1937) 215-223 [[paper](https://gdz.sub.uni-goettingen.de/id/PPN243919689_0177?tify={%22pages%22:[219],%22view%22:%22export%22})]

category: people


Friedrich Kasch

* [personal page](https://www.mathematik.uni-muenchen.de/festschrift/node73.html)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=20292)

* [Wikipedia entry (de)](https://de.wikipedia.org/wiki/Friedrich_Kasch)

## Selected writings

On [[Frobenius extensions]]:

* {#Kasch61} [[Friedrich Kasch]], *Projektive Frobenius-Erweiterungen*, Sitzungsberichte der Heidelberger Akademie der Wissenschaften (1961) [[doi:10.1007/978-3-662-25130-0](https://doi.org/10.1007/978-3-662-25130-0), [pdf](https://epub.ub.uni-muenchen.de/17643/1/17643.pdf)]

category: people


Friedrich Kottler

* [Wikipedia entry](https://en.wikipedia.org/wiki/Friedrich_Kottler) ([de](https://de.wikipedia.org/wiki/Friedrich_Kottler))

## Selected writings

Introducing the perspective that came to be called *[[pre-metric electromagnetism]]*:

* {#Kottler22a} [[Friedrich Kottler]], *Newtonsches Gesetz und Metrik*, Sitzungsber. Akad. Wiss. Wien, Math.-Naturw. Klasse, Abt. IIa, **131** (1922) 1-14 [Engl. transl. by [[David H. Delphenich|Delphenich]]: [pdf](http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/kottler_-_newtons_law_and_metrics.pdf), [[Kottler-NewtonsLawAndMetric.pdf:file]]]

* {#Kottler22b} [[Friedrich Kottler]], *Maxwell'sche Gleichungen und Metrik*, Sitz. Akad. Wien IIa **131** (1922) 119-146 [Engl. transl. by [[David H. Delphenich|Delphenich]]: [pdf](http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/kottler_-_maxwells_equation.pdf), [[Kottler-MaxwellEquationsAndMetric.pdf:file]]]

category: people


Friedrich Levi

* [Wikipedia entry](http://en.wikipedia.org/wiki/Friedrich_Wilhelm_Levi)

category: people


Friedrich Ulmer

* [mathematical genealogy](http://genealogy.math.ndsu.nodak.edu/id.php?id=65958)

## Selected writings

On [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

Introducing the notion of [[locally presentable categories]]:

* [[Pierre Gabriel]], [[Friedrich Ulmer]], *Lokal präsentierbare Kategorien*, Springer LNM **221** (1971) [[doi:10.1007/BFb0059396](https://doi.org/10.1007/BFb0059396), [MR0327863](http://www.ams.org/mathscinet-getitem?mr=0327863)]

On co-[[continuous functors]]:

* [[Friedrich Ulmer]], *On categories of cocontinuous functors. Applications*, Deuxième colloque sur l’algèbre des catégories. Amiens-1975. Résumés des conférences (1975) [[numdam:CTGDC_1975__16_3_217_0](http://www.numdam.org/item/?id=CTGDC_1975__16_3_217_0), [pdf](http://www.numdam.org/item/CTGDC_1975__16_3_217_0.pdf#page=110)]

On [[locally presentable categories]]:

* [[Friedrich Ulmer]]. *Locally presentable categories II* (1975)

On [[bialgebras]] in [[locally presentable categories]]:

* [[Friedrich Ulmer]]. *Bialgebras in locally presentable categories*, University of Wuppertal preprint (1977) [[pdf](/nlab/files/Bialgebras_in_locally_presentable_categories.pdf)]

category:people

[[!redirects F. Ulmer]]
[[!redirects Ulmer]]


Friedrich W. Hehl

* [Institute page](https://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl.html)

* [Wikipedia entry (de)](https://de.wikipedia.org/wiki/Friedrich_W._Hehl)

## Selected writings

On [[pre-metric electromagnetism]]:

* [[Friedrich W. Hehl]], [[Yuri N. Obukhov]], *Foundations of Classical Electrodynamics -- Charge, Flux, and Metric*, Progress in Mathematical Physics **33**, Springer (2003) [[doi:10.1007/978-1-4612-0051-2](https://doi.org/10.1007/978-1-4612-0051-2)]

* {#HehlItinObukhov16} [[Friedrich W. Hehl]], Yakov Itin, Yuri N. Obukhov, *On Kottler's path: origin and evolution of the premetric program in gravity and in electrodynamics*, International Journal of Modern Physics D **25** 11 (2016) 1640016  [[arXiv:1607.06159](https://arxiv.org/abs/1607.06159), [doi:10.1142/S0218271816400162](https://doi.org/10.1142/S0218271816400162)]

## Related entries

* [[electromagnetism]]

category: people

[[!redirects Friedrich Hehl]]


Friedrich Wagemann

* [personal page](https://www.math.sciences.univ-nantes.fr/~wagemann/)

* [Institute page](https://www.univ-nantes.fr/friedrich-wagemann)

## Selected writings

On [[Lie group cohomology]]:

* [[Friedrich Wagemann]], [[Christoph Wockel]], *A Cocycle Model for Topological and Lie Group Cohomology*, Transactions of the American Mathematical Society **367** 3 (2015) 1871-1909 ([arXiv:1110.3304](https://arxiv.org/abs/1110.3304), [jstor:24513021](https://www.jstor.org/stable/24513021))

category: people


Friedrich Wilhelm Bauer

[[!redirects Friedrich Bauer]]
__Friedrich Wilhelm Bauer__ (1931-2019) was a German algebraic topologist. His students include Bernd Günther, Peter Mrozik, Claus Ringel and Wolfgang Metzler.

His contributions include study of stable homotopy theory, homology theories, bordism, categories of chain complexes, Quillen model categories, an original approach to shape theory and especially to  strong shape theory.

His entry in the genealogy project lists his doctoral students, [link](http://genealogy.math.ndsu.nodak.edu/id.php?id=22828)

* de.wikipedia.org/[Friedrich-Wilhelm Bauer](https://de.wikipedia.org/wiki/Friedrich-Wilhelm_Bauer)

category:people


Friedrichs extension

[[Friedrichs extension]] of a non-negative densely defined symmetric operator on a Hilbert space
is a particular (canonically defined, if it exists)
[[self-adjoint extension]] of such an operator.

* [[K. O. Friedrichs]], _Symmetric positive linear differential equations_, Comm. Pure Appl. Math. 11 1958 333–418 [MR100718](http://www.ams.org/mathscinet-getitem?mr=100718) [doi](http://dx.doi.org/10.1002/cpa.3160110306)
* [wikipedia](http://en.wikipedia.org/wiki/Friedrichs_extension)

category: analysis

[[!redirects Friedrichs extensions]]


Friedrichs' system

Related items: [[Friedrichs extension]]

* [[Kurt Otto Friedrichs]], _Symmetric positive linear differential equations_, Comm. Pure Appl. Math. 11 1958 333–418 [MR100718](http://www.ams.org/mathscinet-getitem?mr=100718) [doi](http://dx.doi.org/10.1002/cpa.3160110306) 1958

* A. Ern, J.-L. Guermond, G. Caplain, _An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs' systems_, Commun. in
Part. Diff. Eq. 32 (2007) 317–341 [MR2304151](http://www.ams.org/mathscinet-getitem?mr=2304151) [doi](http://dx.doi.org/10.1080/03605300600718545)
* A. Ern, J.-L. Guermond, _Discontinuous Galerkin methods for Friedrichs' systems. I. General theory_, SIAM J. Numer. Anal., 44 (2006), pp. 753–778 [MR2218968](http://www.ams.org/mathscinet-getitem?mr=2218968) [doi](http://dx.doi.org/10.1137/050624133); _II. Second-order elliptic PDEs_, SIAM J. Numer. Anal., 44 (2006), pp. 2363–2388 [MR2304151](http://www.ams.org/mathscinet-getitem?mr=2304151) [doi](http://dx.doi.org/10.1080/03605300600718545); _III. Multifield theories with partial coercivity._,
SIAM J. Numer. Anal. 46 (2008), no. 2, 776–804 [MR2383211](http://www.ams.org/mathscinet-getitem?mr=2383211) [doi](http://dx.doi.org/10.1137/060664045)

category: analysis

[[!redirects Friedrichs' systems]]
[[!redirects Friedrichs system]]


Frigyes Riesz

* [Wikipedia entry](http://en.wikipedia.org/wiki/Frigyes_Riesz)

## Related entries

* [[Riesz representation theorem]]

category: people


Frm

__$Frm$__ is the [[category]] whose [[objects]] are [[frames]] and whose [[morphisms]] are frame [[homomorphisms]], that is [[lattice]] homomorphisms that preserve [[directed joins]] (and thus all joins).  $Frm$ is a [[subcategory]] of [[Pos]], in fact a [[replete subcategory]] of both [[DistLat]] and [[SupLat]].

The [[opposite category]] of $\mathbf{Frm}$ is the category [[Loc]] of [[locales]]; this is an example of the duality between [[space and quantity]].

## Properties

The category $\mathbf{Frm}$ is [[algebraic category|algebraic]] (see [[Stone Spaces]]).

## Free frames

$Frm$ is given by a [[variety of algebras]], or equivalently by an [[algebraic theory]], so it is an [[equationally presented category]]; however, it requires operations of arbitrarily large arity.  Nevertheless, it is a [[monadic category]] (over [[Set]]), because it has [[free objects]].  Specifically, the __free frame__ on a [[set]] $X$ is the lattice of [[upper subsets]] of the [[free join-semilattice]] on $X$, that is $\mathcal{U}\mathcal{P}_{fin}X$.

(This construction is [[constructive mathematics|constructively valid]], but it does not go through in [[predicative mathematics]], where infinite frames are generally [[proper class|large]] objects.)

Note that free frames are quite different from free [[locales]] (which are [[discrete spaces]] from a localic perspective).

category: category

[[!redirects Frm]]

[[!redirects free frame]]


Frobenioid

A concept in [[arithmetic geometry]] introduced by [[Shinichi Mochizuki]] in the context of his [[inter-universal Teichmüller theory]].

### Related concepts

* [[étale theta function]]

## References

* Wikipedia, _[Frobenioid](http://en.wikipedia.org/wiki/Frobenioid)_

* [[Minhyong Kim]], _[What is a Frobenoid?](http://mathoverflow.net/q/195353/381)_

[[!redirects Frobenioids]]


Frobenius algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Frobenius algebra
* table of contents
{: toc}

## Idea
 {#Idea}

A *Frobenius algebra* is a [[vector space]] equipped with the [[structures]] both of an [[algebra]] and of an a [[coalgebra]] in a compatible way, where the compatibility is different from (more "topological" than) that in a [[bialgebra]]/[[Hopf algebra]]:

The *Frobenius property* on an algebra/coalgebra $A$ states that all ways of using $n$ product operations and $m$ coproduct operations to map $A^{\otimes^{n+1}} \to A^{\otimes^{m+1}}$ are equal.

More generally, Frobenius algebras can be defined [[internalization|internal]] to any [[monoidal category]] (and even in any [[polycategory]]) in which case they are sometimes called *Frobenius monoids*.
(For example a Frobenius monoid in an [[endofunctor|endo]]-[[functor category]] is a *[[Frobenius monad]]*.)

After their original introduction in pure [[algebra]] in the 1930s, Frobenius algebras later attracted much attention for the role they play in [[quantum physics]], in fact for two rather different roles they play there:

* **in [[topological quantum field theory]]** Frobenius algebras encode the structure of [[2-dimensional TQFTs]]:

  Her the underlying vector space of the Frobenius algebra is identified (cf. *[[functorial field theory]]*) with the [[space of quantum states]] over a connected 1-manifold (a [[circle]] for the "[[closed string]]" states or a [[closed interval]] for the "[[open string]]" states), the product and coproduct operations are the [[correlators]] on a [[trinion]] [[cobordism]] ("[[pair of pants]]") read in either direction, and the Frobenius property reflects the [[diffeomorphisms]] between [[cobordisms]] obtained by gluing [[trinions]] along their boundary components.

  Beware that this is similar to but subtly different from the "non-compact" [[2d TQFTs]] commonly known as the "[[topological string]]" (the [[A-model]] and the [[B-model]] of [[mirror symmetry]]-fame) which are *not* described by Frobenius algebras (because here one ex-cludes the "cap" cobordism and thus the [[trace]]-operation hence the finite-dimensionality on the algebra of states) but by "[[Calabi-Yau objects]]" (for more see *[here](TCFT#Definition)* at *[[TCFT]]*.)

  On the other hand, the [[FRS-theorem]] shows that all *[[rational 2d conformal field theories]]* -- hence the compact-target-space sectors of the *physical* [[string]] (such as [[WZW models]] and [[Gepner models]]) -- *are* described by [[Frobenius monoids]] [[internalization|internal]] to [[modular tensor categories]] (MTCs): Here the ambient MTC encodes the local non-topological ("chiral") component of the [[conformal field theory]] (its spaces of [[conformal blocks]]) while the internal Frobenius monoid encodes the remaining topological "[[sewing constraints]]".

Later

* **in [[quantum information theory]]** (such as now in the [[ZX-calculus]]) Frobenius algebras encode aspects of the [[quantum measurement]]-process, including its associated [[wavefunction collapse]] (see [here](quantum+information+theory+via+dagger-compact+categories#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams) at *[[quantum information theory via dagger-compact categories]]*).

  One way to understand how this comes about (not the original way, though) is to observe that in the [[linear type theory]] which is relevant for [[quantum information theory]] the [[reader monad]] and [[coreader comonad]] for a given [[finite set]] $W$ of [[possible worlds|possible]] measurement outcomes merge to a single [[Frobenius monad]] (the *[[quantum reader monad]]*, see there for more) which is in turn identified with the [[writer monad]]/[[cowriter comonad]] for a canonical Frobenius algebra structure on the [[linear span]] of $W$.

Curiously, these two different roles that Frobenius algebras play in [[quantum physics]] are not closely related, in fact they seem somewhat orthogonal to each other.

## Definition

There are a number of equivalent definitions of the concept of Frobenius algebra.

The original definition is an associative algebra with a suitable [[linear form]] on it.

* [As an associative algebra with linear form](#AsAssociativeAlgebraWithLinearForm).

In the context of [[2d TQFT]] what crucially matters is that this is equivalent to an associative algebra structure with a compatible coalgebra structure

* [As an associative algebra with compatible coalgebra structure](#AlgebraCoalgebra).

There are

* [Further equivalent definitions](#FurtherDefinition)

### As associative algebra with coalgebra structure
 {#AlgebraCoalgebra}

+-- {: .num_defn}
###### Definition
A Frobenius algebra in a [[monoidal category]] is a quintuple $(A, \delta, \epsilon, \mu, \eta)$ such that

1. $(A, \mu, \eta)$ is a [[monoid]] with multiplication $\mu:A\otimes A\to A$ and unit $\eta:I\to A$,
1. $(A, \delta, \epsilon)$ is a [[comonoid]] with comultiplication $\delta:A\to A\otimes A$ and counit $\epsilon:A\to I$, and
1. the **Frobenius laws** hold: $(1 \otimes \mu) \circ (\delta \otimes 1) = \delta \circ \mu = (\mu \otimes 1) \circ (1 \otimes \delta)$.
=--

In terms of [[string diagrams]], this definition says:

[[frobenius_algebra.jpg:pic]]

The first line here shows the associative law and left/right unit laws for a [[monoid]].  The second line shows the coassociative law and left/right counit laws for a [[comonoid]].  The third line shows the Frobenius laws.

In fact, although this seems rarely to be remarked, the two Frobenius laws can be replaced by the single axiom $(1 \otimes \mu) \circ (\delta \otimes 1) = (\mu \otimes 1) \circ (1 \otimes \delta)$.  Here is a proof in string diagram notation that this axiom implies $(1 \otimes \mu) \circ (\delta \otimes 1) = \delta \circ \mu$ (taken from [Pastro-Street 2008](#PastroStreet2008)):

[[frobenius_axioms_one_to_two.png:pic]]

Here the first and fifth steps use the counitality property of the comonoid structure, the second and fourth steps use the assumed axiom (once in each direction), and the third step uses coassociativity.

### As associative algebra with linear form
 {#AsAssociativeAlgebraWithLinearForm}

Frobenius algebras were originally formulated in the category [[Vect]] of [[vector spaces]] with the following equivalent definition:

+-- {: .num_defn}
###### Definition

A **Frobenius algebra** is a unital, [[associative algebra]] $(A, \mu, \eta)$ equipped with a linear form $\epsilon : A \rightarrow k$ such that $\epsilon\circ\mu$ is a non-degenerate pairing. I.e. the induced map

\[  u \mapsto (v\mapsto\epsilon\circ\mu(v \otimes u)) \]

is an [[isomorphism]] of $V$ with its [[dual vector space|dual space]] $V^*$. In such a case, $\epsilon$ is called a **Frobenius form**.

=--

From this definition it is easy to see that every Frobenius algebra in [[Vect]] is necessarily finite-dimensional.

### Further definitions
 {#FurtherDefinitions}

There are about a dozen equivalent definitions of a Frobenius algebra. [Ross Street (2004)](#Street2004) lists most of them.

## Types of Frobenius algebras

### Commutative Frobenius algebras ###

We can define 'commutative' Frobenius algebras in any [[symmetric monoidal category]].  Namely, a Frobenius algebra is **commutative** if its associated monoid is commutative --- or equivalently, if its associated comonoid is cocommutative.

### Symmetric Frobenius algebras ###

We can define 'commutative' or 'symmetric' Frobenius algebras in any [[symmetric monoidal category]].  A Frobenius algebra $A$ is **symmetric** if
$$\epsilon \mu \circ S_{A,A} = \epsilon \mu \, $$
where $S_{A,A} : A \otimes A \to A \otimes A$ is the symmetry, and $\epsilon\mu$ is the nondegenerate pairing induced as above from the multiplication and the counit.  Any commutative Frobenius algebra is symmetric, but not conversely: for example the algebra of $n \times n$ matrices with entries in a field, with its usual trace as $\epsilon$, is symmetric but not commutative when $n \gt 1$.

A theorem of [Eilenberg & Nakayama (1955)](#Eilenberg1955) says that in the category [[Vect]] of [[vector spaces]] over a [[field]] $k$, an algebra $A$ can be equipped with the structure of a symmetric Frobenius algebra if (but not only if) it is **[[separable algebra|separable]]**, meaning that for any field $K$ [[field extension|extending]] $k$, $A \otimes_k K$ is a [[semisimple algebra|semisimple]] algebra over $K$.

### Special Frobenius algebras

\begin{definition}
\label{SpecialFrobeniusAlgebras}

The term "special Frobenius algebra" is not used consistently in the literature, but the main point is to require that the product is the [[left inverse]] to the coproduct, possibly up to a non-vanishing element $\beta_A$ in the [[ground field]]:

\[
  \label{ProdLeftInverseToCoprodUpToScaling}
  prod \circ coprod \,=\, \beta_A \cdot id
\]

This condition removes the [[genus of a surface|genus]]-contribution in normal forms of "connected maps", up to rescaling, see [below](#NormalFormAndSpiderTheorem).

But then it is natural to also require the counit to be left inverse to the unit, up to (another) non-vanishing ground field element $\beta_1$:

\[
  \label{CounitLeftInverseToUnitUpTOScaling}
  counit \circ unit \,=\, \beta_1 \cdot id
\]

which also removes the disconnected "vaccum"-contribution, up to scaling.

The two conditions (eq:ProdLeftInverseToCoprodUpToScaling) and (eq:CounitLeftInverseToUnitUpTOScaling) on a Frobenius algebra are taken to be the definition of "special Frobenius" in [Fuchs, Runkel & Schweigert 2002, Def. 3.4 (i)](#FRS-theorem+on+rational+2d+CFT#FuchsRunkelSchweigert02):

\begin{imagefromfile}
    "file_name": "FRS-SpecialFrobeniusDiagram.jpg",
    "width": 400,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 10,
        "left": 10
    }
\end{imagefromfile}

However, other authors take the condition for "special Frobenius" to be the single constraint that

\[
  \label{CoprodLetfInverseToProd}
  prod \circ coprod \,=\, id
\]

(eg. [Coecke & Pavlović 2008, p. 17](quantum+information+theory+via+dagger-compact+categories#CoeckePavlović08), [Fauser 2012, Def. 3.7](https://arxiv.org/abs/1202.6380))

and may instead call (eq:ProdLeftInverseToCoprodUpToScaling) the condition to be *trivially connected*.

Beware that both of these conventions are adhered to by no small number of authors.
\end{definition}

In the category of vector spaces, any element $a$ of an associative unital algebra gives a left multiplication map
$$ \array{
L_a : &A &\to& A \\
      &b &\mapsto& a b
}
$$
which in turn gives a bilinear pairing $g: A \times A \to k$ defined by
$$ g(a,b) = tr(L_a L_b) $$
One can show that the algebra $A$ can be equipped with the structure of a special Frobenius algebra if and only if $g$
is nondegenerate, i.e., if there is an isomorphism $A \to A^*$ given by
$$ a \mapsto g(a, -)  $$
In this case, there is just one way to make $A$ into a special Frobenius algebra, namely by taking the counit to be
$$ \epsilon(a) = tr(L_a) $$
(In any Frobenius algebra, the unit, multiplication and counit determine the comultiplication.)

In fact, all the results of the previous paragraph generalize to Frobenius algebras in any symmetric monoidal category, since the proofs can be done using string diagrams.

An associative unital algebra for which the bilinear pairing $g$ is nondegenerate is called **strongly separable**.  So, any strongly separable algebra becomes a special Frobenius algebra in a unique way.  For more details, see [[separable algebra]] and [Aguiar (2000)](#Aguiar2000).

To get a feeling for some of the concepts we are discussing, an example is helpful.  The group algebra $k[G]$ of a finite group $G$ is always separable but strongly separable if and only if the order of $G$ is invertible in the field $k$.  By the results mentioned, this means that $k[G]$ can always be made into a symmetric Frobenius algebra, but only into a special Frobenius algebra when $|G|$ is invertible in $k$.

To see this, we can check that the group algebra $k[G]$ becomes a symmetric Frobenius algebra if we define the counit $\epsilon: k[G] \to k$ to pick out the coefficient of $1 \in G$:
$$   \epsilon : \sum_{g \in G} a_g \, g \mapsto a_1 \,. $$
But when $|G|$ is invertible in $k$, we can check that $k[G]$ becomes a _special_ symmetric Frobenius algebra if we normalize the counit as follows:
$$  \epsilon : \sum_{g \in G} a_g \, g \mapsto \frac{a_1}{|G|} \, .$$

We should warn the reader that [Rosebrugh et al (2005)](#Rosebrugh2005) call a special Frobenius algebra 'separable'. This usage conflicts with the standard definition of a [[separable algebra]] in the category of vector spaces over a field, so we suggest avoiding it.

### †-Frobenius algebras ###

If a Frobenius algebra lives in a monoidal [[†-category]], $(\delta)^\dagger = \mu$ and $(\epsilon)^\dagger = \eta$, then it is said to be a **†-Frobenius algebra**. These crop up in the theory of 2d [[TQFTs]], and also in the foundations of [[quantum mechanics in terms of dagger-compact categories|quantum theory]].

## Examples
 {#Examples}

\begin{example}
\label{VectorSpaceWithLinearBasis}
\linebreak
For $W \,\in\,$ [[FiniteSets]], the $W$-fold [[direct sum]]
$\mathrm{Q}W \equiv \underset{W}{\oplus} \mathbb{1}$ of the [[ground field]] $\mathbb{1}$ -- with its inherited [[direct sum]]-[[associative algebra]]-[[structure]] -- becomes a [[Frobenius algebra]] by taking the

* [[counit]] to be the canonical [[sum]]-operation;

* [[coproduct]] to be map induced by the [[diagonal]] on indices.

In components, if we choose the canonical [[linear basis]]
$$
  \mathrm{Q}W
  \;\simeq\;
  Span
  \Big(
    \big\{
      \left\vert w \right\rangle
    \big\}_{w \colon W}
  \Big)
$$
then on basis vectors these structure maps are expressed as follows (where "$\delta$" is the [[Kronecker delta]]):

\begin{tikzcd}[sep=0pt]
  \mathbb{K}
  \ar[rr, "{ \mathrm{unit} }"]
  &&
  \mathrm{Q}W
  \\
  1 &\mapsto&
  \sum_{w} \left\vert w \right\rangle
\end{tikzcd}

\begin{tikzcd}[sep=0pt]
  \mathrm{Q}W
  \otimes
  \mathrm{QW}
  \ar[rr, "{ \mathrm{prod} }"]
  &&
  \mathrm{Q}W
  \\
  \left\vert w, \, w' \right\rangle
  &\mapsto&
  \delta_w^{w'}
  \,
  \left\vert w \right\rangle
\end{tikzcd}

\begin{tikzcd}[sep=0pt]
  \mathrm{Q}W
  \ar[rr, "{ \mathrm{coprod} }"]
  &&
  \mathrm{Q}W
  \otimes
  \mathrm{Q}W
  \\
  \left\vert w \right\rangle
  &\;\mapsto\;&
  \left\vert w,\, w \right\rangle
\end{tikzcd}

\begin{tikzcd}[sep=0pt]
  \mathrm{Q}W
  \ar[rr, "{ \mathrm{counit} }"]
  &\;\;&
  \mathbb{K}
  \\
  \left\vert w \right\rangle
  &\mapsto&
  1
\end{tikzcd}

In these components, the Frobenius property is this equation:

\begin{tikzcd}[sep=0pt]
  \mathrm{Q}W
  \otimes
  \mathrm{Q}W
  \ar[
    rrrr,
    "{ \mathrm{id} \,\otimes\, \mathrm{coprod} }"
  ]
  \ar[
    dddd,
    "{
      \mathrm{coprod} \,\otimes\, \mathrm{id}
    }"{description}
   ]
  &&&&
  \mathrm{Q}W
  \otimes
  \mathrm{Q}W
  \otimes
  \mathrm{Q}W
  \ar[
    dddd,
    "{
      \mathrm{prod} \,\otimes\, \mathrm{id}
    }"{description}
  ]
  \\
  &
  \left\vert w,\, w'\right\rangle
  &\mapsto&
  \left\vert w,\, w',\, w' \right\rangle
  \\
  &
  \rotatebox[origin=c]{-90}{$\mapsto$}
  &&
  \rotatebox[origin=c]{-90}{$\mapsto$}
  \\
  &
  \left\vert w,\, w,\, w' \right\rangle
  &\mapsto&
  \delta_{w}^{w'}
  \left\vert w,\, w' \right\rangle
  \\
  \mathrm{Q}W
  \otimes
  \mathrm{Q}W
  \otimes
  \mathrm{W}W
  \ar[
    rrrr,
    "{
      \mathrm{id} \,\otimes\, \mathrm{prod}
    }"{swap}
  ]
  &&&&
  \mathrm{Q}W
  \otimes
  \mathrm{Q}W
\end{tikzcd}

Elementary as this example may be, it embdies some important principles:

For $\mathbb{K} \,=\, \mathbb{C}$ the [[complex numbers]], the corresponding [[Frobenius monad]] $\mathrm{Q}W \otimes (\text{-}) \,\colon\, \mathbb{C}Mod \to \mathbb{C}Mod$ (or rather $\mathrm{Q}W \otimes (\text{-}) \,\colon\, FinDimHilb \to FinDimHilb$) has been proposed [[Coecke & Pavlović 2008](quantum+measurement#CoeckePavlović08)] to reflect aspects of [[quantum measurement]] in the context of [[quantum information via dagger-compact categories]] and is used as such in the *[[zxCalculus]]* (where the Frobenius property is embodied by "spider diagrams"). Various authors discuss the Frobenius algebras $\mathrm{Q}W$ in this context, see the references [there](quantum+information+theory+via+dagger-compact+categories#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams).

From another perspective on the same phenomenon (discussed at *[[quantum circuits via dependent linear types]]*): The $\mathrm{Q}W$-[[writer monad]] underlying the [[Frobenius monad]] $\mathrm{Q}W \otimes (-) \,\colon\, \mathbb{K}Mod \to \mathbb{K}Mod$ is a linear version of the [[reader monad]], as such discussed at *[[quantum reader monad]]*. Regarded as a [[monad in computer science|computational effect]] its [[Kleisli morphisms]] encode [[quantum gates]] with an "indefiniteness"-effect (in the sense of [quantum modal logic](necessity+and+possibility#ModalQuantumLogic)) whose "[handling](monad+in+computer+science#MonadModulesInIdeaSection)" is the process of [[quantum measurement]].

In particular, the [[algebra over a monad|algebras]] (hence also the corresponding [[coalgebra over a comonad|coalgebras]]) over $\mathrm{Q}W \otimes (\text{-})$ the [[dependent linear types|$W$-dependent $\mathbb{K}$-linear types]], namely the $\mathbb{K}$-[[vector bundles]] over $W$, hence the $W$-[[indexed sets]] of $\mathbb{K}$-[[vector spaces]], understood as residual [[spaces of quantum states]] after [[quantum state collapse|collapse]] following the measurement result $w \in W$.
\end{example}

## Properties

### General

+-- {: .num_prop}
###### Proposition
A Frobenius algebra $A$ in a [[monoidal category]] is an [[dual object|object dual]] to itself.
=--

+-- {: .proof}
###### Proof
Let $I$ be the [[monoidal unit]]. To say $A$ is dual to itself means there are maps $e: I \to A \otimes A$ and $p: A \otimes A \to I$ such that the [[triangle identities]] hold. The maps are defined by
$$e = (I \stackrel{\eta}{\to} A \stackrel{\delta}{\to} A \otimes A), \qquad p = (A \otimes A \stackrel{\mu}{\to} A \stackrel{\epsilon}{\to} I)$$
and one of the [[triangle identities]] uses one of the Frobenius laws and unit and counit axioms to derive the following [[commutative diagram]]:

$$\array{
A & \stackrel{1 \otimes \eta}{\to} & A \otimes A & \stackrel{1 \otimes \delta}{\to} & A \otimes A \otimes A \\
 & {}_1\searrow & \downarrow \mathrlap{\mu} & & \downarrow \mathrlap{\mu \otimes 1} \\
 & & A & \overset{\delta}{\to} & A \otimes A \\
 & &  & {}_{1}\searrow & \downarrow \mathrlap{\epsilon \otimes 1} \\
 & & & & A
}
$$

The other [[triangle identity]] uses the other Frobenius law and unit and counit axioms.

=--

As a result, we see that in the monoidal category [[Mod|$Mod_k$]] of [[modules]] over a [[commutative ring]] $k$, Frobenius algebras $A$ considered as modules over $k$ are [[finitely generated module|finitely generated]] and [[projective module|projective]]. This is because $A \otimes_k -$, being [[adjoint functor|adjoint]] to itself, is a [[left adjoint]] and [[left adjoints preserve colimits|therefore preserves all colimits]]. That $A \otimes_k -$ preserves arbitrary small coproducts means $A$ is finitely generated over $k$, and that $A \otimes_k-$ preserves coequalizers means $A$ is projective over $k$.

* Every Frobenius algebra $A$ is a [[quasi-Frobenius algebra]]: projective and injective left (right) modules over $A$ coincide.

* Every Frobenius algebra $A$ is a [[pseudo-Frobenius algebra]]: $A$ is an injective cogenerator in the category of left (right) $A$-modules.

Frobenius algebras are closely connected with [[ambidextrous adjunctions]]. For example, a **[[Frobenius monad]]** on a category $C$ is by definition a Frobenius monoid in the monoidal category of [[endofunctors]] on $C$ (with monoidal product given by endofunctor composition), and if we have a pair of [[adjunctions]] $F \dashv U$ and $U \dashv F$, then $M = U F$ carries a monad structure and a [[comonad]] structure and the Frobenius laws are satisfied, a fact most easily seen by using [[string diagrams]].

### PROPs for Frobenius algebras

Certain kinds of Frobenius algebras have nice [[PROPs]] or [[PRO|PROs]].  The PRO for Frobenius algebras is the monoidal category of planar thick tangles, as noted by [Lauda (2006)](#Lauda2006) and illustrated here:

[[frobenius_algebra.jpg:pic]]

Lauda and Pfeiffer [Lauda (2008)](#Lauda2008) showed that the PROP for _symmetric_ Frobenius algebras is the category of 'topological open strings', since it obeys this extra axiom:

[[symmetric_frobenius_algebra_law.jpg:pic]]

The PROP for _commutative_ Frobenius algebras is [[2Cob]], as noted by many people and formally proved in [Abrams (1996)](#Abrams96).  This means that any commutative Frobenius algebra gives a 2d [[TQFT]].  See [Kock (2006)](#Kock2006) for a history of this subject and [Kock (2004)](#Kock2004) for a detailed introduction.  In 2Cob, the circle is a Frobenius algebra.  The monoid laws look like this:

[[monoid_laws.jpg:pic]]

The comonoid laws look like this:

[[comonoid_laws.jpg:pic]]

The Frobenius laws look like this:

[[frobenius_laws.jpg:pic]]

and the commutative law looks like this:

[[commutative_law.jpg:pic]]

The PROP for _special_ commutative Frobenius algebras is Cospan(FinSet), as proved by Rosebrugh, Sabadini and Walters.  This is worth comparing to the PROP for commutative [[bialgebra|bialgebras]], which is Span(FinSet).  For details, see [Rosebrugh et al (2005)](#Rosebrugh2005), and also [Lack (2004)](#Lack2004).

A special commutative Frobenius algebra gives a 2d TQFT that is insensitive to the genus of a 2-manifold, since in terms of pictures, the 'specialness' axioms $m \circ \delta = 1$ says that

[[special_law.jpg:pic]]

### Classification of 2d TQFT
 {#RelationTo2DTQFT}

[[!include 2d TQFT -- table]]

### Normal form and "Spider theorem"
 {#NormalFormAndSpiderTheorem}

\begin{proposition}
**(normal form for 2d cobordisms)**
\linebreak
Every *[[connected topological space|connected]]* [[cobordism]] with

* $n_{in} \gt 0$ ingoing boundary components

* $n_{out} \gt 0$ outgoing boundary components

* [[genus of a surface|genus]]$\;$ $g$

is equivalent ([[diffeomorphism|diffeomorphic]]) to the [[composition]] of

* $n_{in}-1$ [[trinions]] regarded as $S^1 \times S^1 \to S^1$

* $g$ 2-punctured [[tori]] $S^1 \to S^1$

* $n_{out}-1$ [[trinions]] regared as $S^1 \to S^1 \times S^1$,

\begin{imagefromfile}
    "file_name": "Abrams-2dCobordismNormalForm.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 10,
        "left": 10
    },
    "caption": "(from [Abrams 1996, Fig. 3](#Abrams96))"
\end{imagefromfile}

as shown in the following example for

* $n_{in} = 5$

* $g = 4$

* $n_{out} = 4$

\begin{imagefromfile}
    "file_name": "2dCObordismNormalForm.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 10,
        "left": 10
    },
    "caption": "(from [Kock 2004, p. 64](#Kock2004))"
\end{imagefromfile}

\end{proposition}
This is discussed in [Abrams 1996, Prop. 12](#Abrams96); [Kock 2004, §1.4.16](#Kock2004).

By the [relation](#RelationTo2DTQFT) between 2d cobordism and Frobenius algebras this means equivalently that:

\begin{corollary}\label{NormalFormForFrobeniusMaps}
**(normal form for Frobenius maps)**
\linebreak
Given a Frobenius algebra

$$
  \big(
    A
    ,\,
    unit
    ,\,
    counit
    ,\,
    prod
    ,\,
    coprod
  \big)
$$

then every linear map of the form

$$
  A^{\otimes^{n_{in}}}
  \longrightarrow
  A^{\otimes^{n_{out}}}
$$

which is

1. entirely the composite of the algebra's structure maps, ie. of the (co)unit and the (co)product,

1. *connected*, in that it does not decompose as a tensor product of such morphisms,

is equal to the composite, in this order, of

* $n_{in}-1$ copies of $prod$

* some number ($g$) of copies of $prod \circ coprod$

* $n_{out}-1$ copies of $coprod$.

\end{corollary}

\begin{corollary}\label{SpiderTheorem}
**(Spider theorem)**
\linebreak
If the Frobenius algebra in Cor. \ref{NormalFormForFrobeniusMaps} is *special* in that $prod \circ coprod = id$ (eq:CoprodLetfInverseToProd) then every linear map of the form
$$
  A^{\otimes^{n_{in}}}
  \longrightarrow
  A^{\otimes^{n_{out}}}
$$

which is

1. entirely the composite of the algebra's structure maps, ie. of the (co)unit and the (co)product,

1. *connected* in that it does not decompose as a tensor product of such morphisms,

is equal to the composite, in this order, of

* $n_{in}-1$ copies of $prod$

* $n_{out}-1$ copies of $coprod$.

\end{corollary}

\begin{imagefromfile}
    "file_name": "FrobNormalFormAsSpiderRule.jpg",
    "float": "right",
    "width": 500,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Cocke & Duncan 2011](quantum+information+theory+via+dagger-compact+categories#CoeckeDuncan11)"
\end{imagefromfile}

The "spider theorem" (Cor. \ref{SpiderTheorem}) is called this way in [Coecke & Duncan 2008, Thm. 1](quantum+information+theory+via+dagger-compact+categories#CoeckeDuncan08) (it appears unnamed also in [Coecke & Paquette 2008, p. 6](quantum+information+theory+via+dagger-compact+categories#CoeckePaquette08)) and shares its name with the corresponding "spider diagrams" used in the [[ZX-calculus]] for [[quantum computation]] (where the Frobenius algebras that appear are those that realize the [[quantum reader monad]] as a [[Frobenius monad|Frobenius]] [[writer monad]]/[[cowriter comonad]]).

### Frobenius algebras in polycategories

In fact, Frobenius algebras can be defined in any [[polycategory]], and hence in any [[linearly distributive category]].  The essential point is that the monoidal structure used for the monoid structure could be different from the monoidal structure used for the comonoid structure, i.e. we could have $\mu:A\otimes A \to A$ but $\delta :A \to A \parr A$.  The compatibility between $\otimes$ and $\parr$ in a linearly distributive category (or between their "[[multicategory|multicategorical]]" analogues in a polycategory) is precisely what is required to write down the composites involved in the Frobenius laws.  For instance, we can have

$$ A\otimes A
\xrightarrow{1_A \otimes \delta}
A \otimes (A \parr A)
\xrightarrow{lin-distrib}
(A \otimes A) \parr A
\xrightarrow{\mu \parr 1_A}
A\parr A
$$

and one of the Frobenius laws says that this composite is equal to

$$ A \otimes A \xrightarrow{\mu} A \xrightarrow{\delta} A\parr A. $$

This is analogous to how a [[bimonoid]] can be defined in any [[duoidal category]].  In fact, it is a sort of [[microcosm principle]]; it is shown in [(Egger2010)](#Egger2010) that Frobenius monoids in the linearly distributive category [[Sup]] are precisely [[star-autonomous category|*-autonomous]] cocomplete [[posets]] (and hence, in particular, linearly distributive).

In polycategorical language we can give another [[unbiased]] definition of a  commutative Frobenius monoid: it is equipped with exactly one morphism $\overset{n}{\overbrace{(A,A,\dots,A)}} \to \overset{m}{\overbrace{(A,A,\dots,A)}}$ of each possible (two-sided) arity, such that any (symmetric) polycategorical composite of two such morphisms is equal to another such.  The monoid structure consists of the morphisms of arity $(2,1)$ and $(0,1)$, while the comonoid structure is the morphisms of arity $(1,2)$ and $(1,0)$, and the Frobenius relations say that three ways to compose these to produce a morphism of arity $(2,2)$ are equal.  (The morphism of arity $(0,0)$ is the composite $\epsilon\eta$; no axiom is required on it, because in a polycategory there is no other morphism $()\to ()$ to compare it to.)  In other words, the free symmetric polycategory containing a commutative Frobenius monoid is the [[terminal object|terminal]]  symmetric polycategory.  In this way Frobenius algebras are to polycategories in the same way that monoids are to multicategories.

+--{: .query}
[[Mike Shulman]]: I have not carefully checked the above statement, but it seems that the Frobenius laws should suffice to manipulate any such composite into any other.  Personal communications from other people who should know are in agreement.
=--

## Related concepts

* [[Calabi-Yau algebra]]

* [[hypergraph category]]

* [[Frobenius pseudomonoid]]

[[!include reader-writer (co)monads -- table]]

## References ##

Frobenius algebras were introduced by [[Brauer]] and Nesbitt and were named after [[Ferdinand Frobenius]].

See for instance

* {#Eilenberg1955} [[Samuel Eilenberg]], [[Tadasi Nakayama]], _On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings_, _Nagoya Math. J._ **9** (1955) 1-16 [[doi:10.1017/S0027763000023229](https://doi.org/10.1017/S0027763000023229)]

* [[Marcelo Aguiar]] (2000), _A note on strongly separable algebras_, Boletín de la Academia Nacional de Ciencias (Córdoba, Argentina), special issue in honor of Orlando Villamayor, **65**, 51--60.  ([pdf](http://www.math.cornell.edu/~maguiar/strongly.pdf))
{#Aguiar2000}

* [[Andrew Baker]], _Frobenius algebras_ 2010 ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/Frobenius-talk.pdf))

On the role of Frobenius algebras in [[2d TQFT]]:

* {#Abrams96} [[Lowell Abrams]], *Two-dimensional topological quantum field theories and Frobenius algebra*, Jour. Knot. Theory and its Ramifications **5**, 569-587 (1996) [[doi:10.1142/S0218216596000333](https://doi.org/10.1142/S0218216596000333), [ps](http://home.gwu.edu/~labrams/docs/tqft.ps)]

* {#BaezTWF} [[John Baez]], This Week's Finds in Mathematical Physics, [week268](http://math.ucr.edu/home/baez/week268.html) and [week299](http://math.ucr.edu/home/baez/week299.html).

* {#Kock2004} [[Joachim Kock]], *Frobenius Algebras and 2d Topological Quantum Field Theories*, Cambridge U. Press (2004) [[doi:10.1017/CBO9780511615443](https://www.cambridge.org/core/books/frobenius-algebras-and-2d-topological-quantum-field-theories/A6438118DFADFD27175779F1FC0FF7CB), [webpage](http://mat.uab.cat/~kock/TQFT.html), [course notes pdf](http://mat.uab.es/~kock/TQFT/FS.pdf), [[Kock-FrobAlgTQFT-short.pdf:file]]]

* {#Kock2006} [[Joachim Kock]], Remarks on the history of the Frobenius equation (2006) [[web](http://mat.uab.es/~kock/TQFT.html#history)]

* {#Lauda2006} [[Aaron Lauda]], Frobenius algebras and ambidextrous adjunctions, *Theory and Applications of Categories* **16** (2006) 84-122 [[tac:16-04](http://tac.mta.ca/tac/volumes/16/4/16-04abs.html), [arXiv:math/0502550](http://arxiv.org/abs/math/0502550)]

  > (relating to [[ambidextrous adjunctions]] and [[Frobenius monads]])

* {#Lauda2008} [[Aaron Lauda]], [[Hendryk Pfeiffer]], Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, _Topology Appl._ **155** (2005) 623-666 [[arXiv:0510664](http://arxiv.org/abs/math/0510664), [doi:10.1016/j.topol.2007.11.005](https://doi.org/10.1016/j.topol.2007.11.005)]

For applications in proof theory of classical and [[linear logic]] or linguistics:

* [[Martin Hyland]],  _Abstract Interpretation of Proofs: Classical Propositional Calculus_, pp. 6-21 in Marcinkowski, Tarlecki (eds.),  _Computer Science Logic (CSL 2004)_, LNCS **3210** Springer Heidelberg 2004. ([preprint](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2004/aap04.pdf))

* [[Richard Garner]], _Three investigations into linear logic_ , PhD report  Cambridge 2006. ([pdf](http://comp.mq.edu.au/~rgarner/Thesis/Smith-Knight-Essay.pdf))

* D. Kartsaklis, M. Sadrzadeh, S. Pulman,  [[Bob Coecke]], _Reasoning about meaning in natural language with compact closed categories and Frobenius algebras_ (2014). ([arXiv:1401.5980](https://arxiv.org/abs/1401.5980))

Frobenius algebras in linearly distributive categories are discussed in

* [[Jeff Egger]], *The Frobenius relations meet linear distributivity*, 2010 [TAC](http://tac.mta.ca/tac/volumes/24/2/24-02abs.html)
 {#Egger2010}

See also

* Aurelio Carboni (1991), Matrices, relations, and group representations, _Journal of Algebra_ **136** 2, 497--529.  ([pdf](https://www.sciencedirect.com/science/article/pii/002186939190057F))

* [[Bertfried Fauser]], _Some Graphical Aspects of Frobenius Structures_,  preprint (2012).  [arXiv:1202.6380](http://arxiv.org/pdf/1202.6380v1)

* {#Lack2004} [[Stephen Lack]] (2004), Composing PROPs, _Theory and Applications of Categories_ **13**, 147--163.  ([web](http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html))

* [[F. W. Lawvere]], _Ordinal Sums and Equational Doctrines_, pp. 141-155 in Eckmann (ed.), _Seminar on Triples and Categorical Homology Theory_, LNM **80** Springer Heidelberg 1969. ([TAC Reprint of vol. 80](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf))

* {#PastroStreet2008} Craig Pastro and [[Ross Street]], _Weak Hopf monoids in braided monoidal categories_, 2008, [arXiv:0801.4067](https://arxiv.org/abs/0801.4067)

* {#Rosebrugh2005} R. Rosebrugh, N. Sabadini and R.F.C. Walters (2005), Generic commutative separable algebras and cospans of graphs, _Theory and Applications of Categories_ **15** (Proceedings of CT2004), 164--177.  ([web](http://www.tac.mta.ca/tac/volumes/15/6/15-06abs.html))

* {#Street2004} [[Ross Street]] (2004), Frobenius monads and pseudomonoids, _J. Math. Phys._ **45**.  ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.91.2686))

* R. F. C. Walters, R. J. Wood, _Frobenius objects in Cartesian cicategories_, TAC **20** no. 3 (2008) 25--47. ([pdf](http://www.tac.mta.ca/tac/volumes/20/3/20-03.pdf))

category: algebra

[[!redirects Frobenius algebra]]
[[!redirects Frobenius algebras]]
[[!redirects Frobenius monoid]]
[[!redirects Frobenius monoids]]


Frobenius category

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One definition says (where?) that:
A __Frobenius category__ is a [[Quillen exact category]] which has enough [[injective object|injectives]] and enough [[projective object|projectives]] and where the class of projectives coincides with the class of injectives. The [[stable category]] of a Frobenius category is canonically a [[triangulated category]]. If a triangulated category $T$ is triangle equivalent to the stable category of a Frobenius category, then we say that $T$ is _algebraic_.

Another (?) definition is given in [Etingof & Ostrik 2021, §1.3](#EtingofOstrik21).

## References

* {#EtingofOstrik21} [[Pavel Etingof]], [[Victor Ostrik]], *On the Frobenius functor for symmetric tensor categories in positive characteristic*, J. für die reine und angewandte Mathematik **773** (2021) 165–198 $[$[arXiv:1912.12947](https://arxiv.org/abs/1912.12947)$]$

In view of [[Deligne's theorem on tensor categories]]:

* {#CoulembierEtingofOstrik21} [[Kevin Coulembier]], [[Pavel Etingof]], [[Victor Ostrik]], _On Frobenius exact symmetric tensor categories_ $[$[arXiv:2107.02372](https://arxiv.org/abs/2107.02372)$]$

[[!redirects Frobenius categories]]

[[!redirects Frobenius exact category]]
[[!redirects Frobenius exact categories]]


Frobenius functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

\begin{definition}\label{FrobeniusFunctor}
**(Frobenius functor)**
\linebreak
An [[adjoint functor|adjoint pair]]  of [[functors]]

$$
  (F\dashv G) \;\colon\; D \longrightarrow C
$$

is called a _Frobenius pair_ if $F$ is not only a [[left adjoint]] to $G$ but also a [[right adjoint]] to $G$, hence if we have an [[ambidextrous adjunction]], i.e. an [[adjoint triple]] of the form

$$
  (F \dashv G \dashv F) \;\colon\; D \longrightarrow C
  \,.
$$

This situation was called a *[[strongly adjoint pair]]* by [Morita 1965](#Morita65), but the [Caenepeel, Militaru & Zhu 1997](#CaenepeelMilitaruZhu97) (followed by [CGN 1998](#CGN98)) suggested to refer to such $F$ as *Frobenius functors*.
\end{definition}

\begin{remark}\label{Terminology}
**(terminology)**
\linebreak
The terminology "Frobenius functor" (Def. \ref{FrobeniusFunctor}) is motivated by the observation (Thm. 5.1 there, see [below](#FrobeniusExtensions)) of [Morita 1965](#Morita65) (who instead speaks of "[[strongly adjoint pairs]]"!) that the functor of [[restriction of scalars]] $f^\ast$ along a [[homomorphism]] $f$ of [[rings]] has coinciding [[left adjoint]] ([[extension of scalars]]) and [[right adjoint]] ([[coextension of scalars]]) iff $f$ is a [[ring extension]] which is a "Frobenius extension" in the sense of [Kasch 1961](#Kasch61).

The suggestion to therefore refer to Morita's "strongly adjoint pairs" as "Frobenius functors" is due to [Caenepeel, Militaru & Zhu 1997](#CaenepeelMilitaruZhu97), seconded by [CGN 1998](#CGN98).

Beware of the independent terminology of "*[[Frobenius monoidal functor]]*" which does *not* refer to Frobenius functors that are also [[monoidal functor|monoidal]]. (Similarly, one should *not* refer to Kasch's *Frobenius extensions* as "[[Frobenius morphisms]]"!)

On the other hand, the ([[comonad|co]])[[monads]] [induced](monad#RelationBetweenAdjunctionsAndMonads) by Frobenius adjunctions are the *[[Frobenius monads]]*, see the discussion [there](Frobenius+monad#Properties).
\end{remark}

## Examples

\begin{example}
\label{FrobeniusExtensions}
[Morita 1965](#Morita65) proved that the [[extension of scalars]] functor for a [[morphism]] of [[rings]] $f:R\to S$ is Frobenius iff the morphism $f$ itself is a Frobenius extension  in the sense of ([Kasch 1961](#Kasch61)), that is: ${}_R S$ is finitely generated [[projective object|projective]] and ${}_S S_R\cong Hom_R({}_R S, {}_R R)$ as $R-S$-[[bimodule]]s.

  This is in the spirit of the finite-dimensional duality coded e.g. in the notion of [[Frobenius algebra]].
\end{example}

## References

The concept originates under the name "[[strongly adjoint pair]]" in:

* {#Morita65} [[Kiiti Morita]], *Adjoint pairs of functors and Frobenius extensions*, Science Reports of the Tokyo Kyoiku Daigaku, Section A **9** 202/208 (1965) 40-71 [[jstor:43698658](https://www.jstor.org/stable/43698658)]

in application to the notion of "Frobenius extension" due to

* {#Kasch61} [[Friedrich Kasch]], *Projektive Frobenius-Erweiterungen*, Sitzungsberichte der Heidelberger Akademie der Wissenschaften (1961) [[doi:10.1007/978-3-662-25130-0](https://doi.org/10.1007/978-3-662-25130-0), [pdf](https://epub.ub.uni-muenchen.de/17643/1/17643.pdf)]

The terminology "Frobenius functor" for this situation is due to:

* {#CaenepeelMilitaruZhu97}  [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], *Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties*, Trans. Amer. Math. Soc. **349** (1997) 4311-4342 [[1997-349-11/S0002-9947-97-02004-7](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7), [pdf](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7/S0002-9947-97-02004-7.pdf)]

* {#CGN98} F. Castaño Iglesias, [[José Gómez-Torrecillas]], C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__ 10  (1998) 4879-4900 [[doi:10.1080/00927879908826735](https://doi.org/10.1080/00927879908826735)]

Further discussion:

* {#CaenepeelDeGrooMilitaru02} [[Stefaan Caenepeel]], E. De Groot, [[Gigel Militaru]], *Frobenius Functors of the second kind*, Comm. Algebra **30** (2002) 5359-5391 [[arXiv:math/0106109](https://arxiv.org/abs/math/0106109), [doi:10.1081/AGB-120015657](https://doi.org/10.1081/AGB-120015657)]

* [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], _Frobenius and separable functors for generalized module categories and nonlinear equations_, Springer Lec. Notes in Math. __1787__ (2002) [[gBooks](http://books.google.hr/books?id=6erTEpKIUaYC&lpg=PA89&ots=-TGKqXRstB&dq=Frobenius%20functors&hl=en&pg=PA89#v=onepage&q&f=false)]

[[!redirects Frobenious functor]]
[[!redirects Frobenious functors]]
[[!redirects Frobenius functors]]

[[!redirects Frobenius extension]]
[[!redirects Frobenius extensions]]


Frobenius manifold

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[quantum cohomology ring]]

## References

* [[Boris Dubrovin]], *Geometry of 2d topological field theories*, in: *Integrable Systems and Quantum Groups*, Lecture Notes in Mathematics **1620** (1996) [[arXiv:hep-th/9407018](https://arxiv.org/abs/hep-th/9407018), [doi:10.1007/BFb0094793](https://doi.org/10.1007/BFb0094793)]

* [[Boris Dubrovin]], *Geometry and Analytic Theory of Frobenius Manifolds*, Extra Volume ICM II (1998) 315–326 [[arXiv:math/9807034](https://arxiv.org/abs/math/9807034)]

See also:

* Wikipedia, _[Frobenius manifold](http://en.wikipedia.org/wiki/Frobenius_manifold)_

[[!redirects Frobenius manifolds]]


Frobenius monad

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Frobenius monad_ [[Lawvere 1969](#Lawvere69)] is a _[[monad]]_ which as a [[monoid]] in [[endofunctors]] is a [[Frobenius monoid]].

## Properties
 {#Properties}

The monad [induced](monad#RelationBetweenAdjunctionsAndMonads) from an [[ambidextrous adjunction]] is Frobenius. The converse is also true:

\begin{prop}
\label{FrobMonadsHaveAmbidextrousEMAdjunctions}
If $M$ is a Frobenius monad on a category $C$, then the usual [[free-forgetful adjunction]] $F \dashv U \colon Alg_M \to C$ on its [[Eilenberg-Moore category]] is an [[ambidextrous adjunction]] whose associated Frobenius monad is isomorphic to $M$.
\end{prop}
(cf. [Street (2004), Section II](#Street04))
\begin{proof}
In general (see [this Prop.](adjoint+monad#IsomorphismOfEMCategories) at *[[adjoint monad]]*), if a monad $M$ admits a [[right adjoint]] $K$, then $K$ carries a [[comonad]] [[structure]] [[mate|mated]] to the monad structure of $M$, and there is an [[adjoint equivalence]] $Alg_M \simeq CoAlg_K$ between their [[Eilenberg-Moore categories]] (considered as categories over $C$, via the usual [[forgetful functors]]).

Now if $M$ is Frobenius, then $M \dashv M$ and the comonad structure mated to the monad is indeed the given comonad structure of $M$ (a proof is given at *[[Frobenius algebra]]* (see [there](Frobenius+algebra#general)). Hence the [[left adjoint]] $C \to Alg_M$ to the [[forgetful functor]] may be identified with the [[right adjoint]] $C \to CoAlg_M$ of the forgetful functor, each being unique (up to isomorphism) lifts of $M \colon C \to C$ through the forgetful functors.
\end{proof}

## Examples
 {#Examples}

\begin{example}
\end{example}

\begin{example}\label{FrobeniusWriterMonad}
**(Frobenius writer monad)**
\linebreak
  For $A$ a [[Frobenius algebra]] (over some [[ground field]] $\mathbb{K}$) the [[writer monad]]/[[cowriter comonad]] $A \otimes (-) \,\colon\,$ is canonically a Frobenius monad, with (co)product and (co)unit induced from the corresponding operations on $A$.

[[!include reader-writer (co)monads -- table]]
\end{example}

\begin{example}\label{LinearCoReaderMonad}
**(linear (co)reader monad)**
\linebreak
For $W$ a [[finite set]] and $\mathbb{K}$ any [[ground field]], the [[reader monad]] $\bigcirc_W \;\colon\; Vect_{\mathbb{K}} \to Vect_{\mathbb{K}}$ on $\mathbb{K}$-[[VectorSpaces]] is Frobenius and [[isomorphism|isomorphic]] (see [there](function+monad#QuantumReaderMonadIsSpecialFrobeniusWriterMonad) for details) with the corresponding [[comonad]] being the linear [[cowriter comonad]], this reflecting that [[finite products]] of vector spaces are [[biproducts]], namely [[direct sums]].

Finally, this Frobenius monad $\bigcirc_W \,\simeq\, \bigstar_W \,\colon\, Vect \to Vect$ is isomorphic to the Frobenius [[writer monad]] (of Exp. \ref{FrobeniusWriterMonad}) corresponding to the [[Frobenius algebra]] $\mathrm{Q}W \,\equiv\,\oplus_W \mathbb{K}$ (see [there](Frobenius+algebra#VectorSpaceWithLinearBasis)).

As such, it has been proposed [[Coecke & Pavlović 2008](quantum+measurement#CoeckePavlović08)] to reflect aspects of [[quantum measurement]] in the context of [[quantum information via dagger-compact categories]] and is used as such in the *[[zxCalculus]]* (where the Frobenius property is embodied by "spider diagrams"). Various authors discuss the Frobenius monads in this context, see the references [there](quantum+information+theory+via+dagger-compact+categories#MeasurementReferencesQuantumInformationTheoryViaStringDiagrams).
\end{example}

## Related concepts

* [[Wirthmüller context]]

* [[infinitesimal cohesion]]

## References

The notion of Frobenius monads appears briefly in

* {#Lawvere69} [[William Lawvere]], pp. 151 of: *Ordinal sums and equational doctrines*, in *[[Seminar on Triples and Categorical Homology Theory]]* LNM **80** (1969) 141-155 [[doi:10.1007/BFb0083085](https://doi.org/10.1007/BFb0083085)]

The relation of Frobenius monads to [[ambidextrous adjunctions]]:

in [[Cat]]

* {#Street04} [[Ross Street]], *Frobenius monads and pseudomonoids*, J. Math. Phys. **45** 3930 (2004) [[doi:10.1063/1.1788852](https://doi.org/10.1063/1.1788852)]

and in general [[2-categories]]:

* {#Lauda05} [[Aaron Lauda]], *Frobenius algebras and ambidextrous adjunctions*, Theory and Applications of Categories **16** 4 (2006) 84-122 [[arXiv:math/0502550](http://arxiv.org/abs/math/0502550), [tac:16-04](http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html)]

  > (motivated by [[2d TQFT]])

Discussion of [[ambidextrous adjunctions]] between [[abelian categories]] under the name *[[Frobenius morphisms]]*:

* F. Castaño Iglesias, [[José Gómez Torrecillas]], C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__ 10  (1998) 4879-4900 [[doi:10.1080/00927879908826735](https://doi.org/10.1080/00927879908826735)]

On Frobenius ([[cowriter comonad|co]])[[writer monads]] induced form [[Frobenius monoids]] with special attiotion to their [[strong monad|strength]] (following [Wolff 1973](action+monad#Wolff73)):

* [[Martti Karvonen]], *Frobenius algebras in functor categories*, Oxford (2014) [[pdf](https://www.cs.ox.ac.uk/people/aleks.kissinger/theses/bob/Karvonen.pdf), [[Karvonen-FrobeniusMonads.pdf:file]]]

and analogous discussion in [[dagger-categories]]:

* [[Chris Heunen]], [[Martti Karvonen]], *Monads on dagger categories*, Theory and Applications of Categories **31** 35 (2016) 1016-1043 [[arXiv:1602.04324](https://arxiv.org/abs/1602.04324), [tac:31-35](http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html)]

[[!redirects Frobenius monads]]


Frobenius monoid > history

< [[Frobenius algebra]]


Frobenius monoidal functor

[[!redirects Frobenius monoidal functors]]
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###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[functor]] $F : C \to D$ between [[monoidal categories]] which is both a [[lax monoidal functor]] and an [[oplax monoidal functor]] is called **Frobenius** if the structure morphisms for the lax monoidal structure

$$
  \nabla_{x,y} : F(x)\otimes F(y) \to F(x \otimes y)
$$

and for the oplax monoidal structure

$$
  \Delta_{x,y} : F(x \otimes y) \to F(x) \otimes F(y)
$$

satisfy the axioms of a [[Frobenius algebra]] in that (in [[string diagram]] notation) for all objects $x,y,z$ in $C$ we have

$$
  \left(
    \array{
      F(x) &&&& F(y \otimes z)
      \\
      \downarrow &&& \swarrow & \downarrow
      \\
     F(x) && F(y)& & F(z)
     \\
     \downarrow & \swarrow &&& \downarrow
     \\
     F(x \otimes y) &&&& F(z)
    }
  \right)
  =
  \left(
  \array{
    F(x) &&&& F(y \otimes z)
    \\
    & \searrow && \swarrow
    \\
    && F(x \otimes y \otimes z)
    \\
    & \swarrow && \searrow
    \\
    F(x \otimes y) &&&& F(y)
  }
  \right)
$$

and

$$
  \left(
    \array{
      F(x \otimes y) &&&& F(z)
      \\
      \downarrow & \searrow && & \downarrow
      \\
     F(x) && F(y)  & & F(z)
     \\
     \downarrow & && \searrow & \downarrow
     \\
     F(x ) &&&& F(y \otimes z)
    }
  \right)
  =
  \left(
  \array{
    F(x \otimes y) &&&& F(z)
    \\
    & \searrow && \swarrow
    \\
    && F(x \otimes y \otimes z)
    \\
    & \swarrow && \searrow
    \\
    F(x) &&&& F(y \otimes z)
  }
  \right)
$$

## Examples

The [[Moore complex]] functor

$$
  C : sAb \to Ch_\bullet^+
$$

from abelian [[simplicial group]]s to connective [[chain complex]]es is Frobenius, as is the normalzed chains complex functor

$$
  N : sAb \to Ch_\bullet^+
  \,.
$$

For more on this see [[monoidal Dold-Kan correspondence]].

## Generalizations

Frobenius monoidal functors are a special case of Frobenius *linear* functors between [[linearly distributive categories]].

## Related concepts

* [[monoidal functor]]

* [[strong monoidal functor]]

* [[lax monoidal functor]]

* [[oplax monoidal functor]]

* [[bilax monoidal functor]]

* **Frobenius monoidal functor**

Beware of the *un*-related terminoilogy:

* [[Frobenius functor]]

## References

Equation (3.26), (3.27) in [p. 81](http://www.math.tamu.edu/~maguiar/a.pdf) of

* [[Marcelo Aguiar]], [[Swapneel Mahajan]], _[[Monoidal Functors, Species and Hopf Algebras]]_
{#AguiarMahajan}
* M. B. McCurdy, [[R. Street]], _What separable Frobenius monoidal functors preserve_, [arxiv/0904.3449](http://arxiv.org/abs/0904.3449) and Cahiers TGDC, 51 (2010)p. 29 - 50.


Frobenius morphism

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###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[number theory]], [[Galois theory]] and [[arithmetic geometry]] in [[prime number|prime]] [[characteristic]] $p$, the _Frobenius morphism_ is the [[endomorphism]] acting on [[algebras]], [[function algebras]], [[structure sheaves]] etc., which takes each [[ring]]/[[associative algebra|algebra]]-element $x$ to its $p$th power

$$
  x^p = \underbrace{x \cdot x \cdots x}_{p \; factors}
  \;.
$$

It is precisely in [[positive characteristic]] $p$ that this operation is indeed an algebra [[homomorphism]] ("[[freshman dream arithmetic]]").

The Frobenius map is the shadow of the [[power operations]] in [[multiplicative cohomology theory]]/[[higher algebra]] ([Lurie, remark 2.2.7](#Lurie)).

The presence of the Frobenius endomorphism in characteristic $p$ is a fundamental property in [[arithmetic geometry]] that controls many of its deep aspects. Notably [[zeta functions]] are typically expressed in terms of the [[action]] of the Frobenius endomorphisms on [[cohomology groups]] and so it features prominently for instance in the [[Weil conjectures]].

In [[Borger's absolute geometry]] lifts of Frobenius endomorphisms through [[base change]] for all primes at once -- in the sense of _[Lambda-ring structure](Lambda-ring#LambdaRingByFrobeniusLifts)_ -- is interpreted as encoding [[descent]] data from traditional [[arithmetic geometry]] over [[Spec(Z)]] down to the "absolute" geometry over "[[F1]]". See at _[Borger's absolute geometry -- Motivation](Borger%27s+absolute+geometry#Motivation)_ for more on this.

## Definition

> this entry may need attention

## For fields

Let $k$ be a [[field]] of positive [[characteristic]] $p$. The **Frobenius morphism** is an [[endomorphism]] of the field $F \colon k \to k$ defined by

$$
  F(a) \coloneqq  a^p
  \,.
$$

Notice that this is indeed a [[homomorphism]] of fields: the identity $(a b)^p=a^p b^p$ evidently holds for all $a,b\in k$ and the [[characteristic]] of the field is used to show $(a+b)^p=a^p+b^p$.

### Of schemes

Suppose $(X,\mathcal{O}_X)$ is an $S$-[[scheme]] where $S$ is a [[scheme]] over $k$. The **absolute Frobenius** is the map $F^{ab}:(X,\mathcal{O}_X)\to (X,\mathcal{O}_X)$ which is the identity on the [[topological space]] $X$ and on the structure sheaves $F_*:\mathcal{O}_X\to \mathcal{O}_X$ is the $p$-th power map. This is not a map of $S$-schemes in general since it doesn't respect the structure of $X$ as an $S$-scheme, i.e. the diagram:

$\displaystyle \begin{matrix} X & \stackrel{F^{ab}}{\to} & X \\
\downarrow &  & \downarrow \\
S & \stackrel{F^{ab}}{\to} & S \end{matrix}$,

so in order for the map to be an $S$-scheme morphism, $F^{ab}$ must be the identity on $S$, i.e. $S=Spec(\mathbb{F}_p)$.

Now we can form the [[fiber product]] using this square: $X^{(p)}:=X\times_{S} S$. By the universal property of [[pullback]]s there is a map $F^{rel}:X\to X^{(p)}$ so that the composition $X\to X^{(p)}\to X$ is $F^{ab}$. This is called the **relative Frobenius**. By construction the relative Frobenius is a map of $S$-schemes.

### For sheaves on $C Ring ^{op}$

Let $p$ be a prime number, let $k$ be a field of characteristic $p$. For a $k$-ring $A$ we define

$$f_A:
\begin{cases}
A\to A
\\
x\mapsto x^p
\end{cases}$$

The $k$-ring obtained from $A$ by scalar restriction along $f_k:k\to k$ is denoted by $A_{f}$.

The $k$-ring obtained from $A$ by scalar extension along $f_k:k\to k$ is denoted by $A^{(p)}:=A\otimes_{k,f} k$.

There are $k$-ring morphisms $f_A: A\to A_f$ and $F_A:\begin{cases}
A^{(p)}\to A
\\
x\otimes \lambda\mapsto x^p \lambda
\end{cases}$.

For a $k$-functor $X$ we define $X^{(p)}:X\otimes_{k,f_k} k$ which satisfies $X^{(p)}(R)=X(R_f)$. The _Frobenius morphism_ for $X$ is the transformation of $k$-functors defined by

$$F_X:
\begin{cases}

X\to X^{(p)}
\\
X(f_R):X(R)\to X(R_f)
\end{cases}$$

If $X$ is a $k$-scheme $X^{(p)}$ is a $k$-scheme, too.

Since the completion functor ${}^\hat\;:Sch_k\to fSch_k$ commutes with the above constructions the Frobenius morphism can be defined for [[formal scheme|formal k-schemes]], too.

### In terms of symmetric products
 {#InTermsOfSymmetricProducts}

We give here another characterization of the [[Frobenius morphism]] in terms of symmetric products.

Let $p$ be a prime number, let $k$ be a field of [[characteristic]] $p$, let $V$ be a $k$-vector space, let $\otimes^p V$ denote the $p$-fold tensor power of $V$, let $TS^p V$ denote the subspace of symmetric tensors, yielding the [[symmetric algebra]]. Then we have the symmetrization operator

$$s_V:
\begin{cases}
\otimes^p V\to TS^p V
\\
a_1\otimes\cdots\otimes a_n\mapsto \Sigma_{\sigma\in S_p}a_{\sigma(1)}\otimes\cdots\otimes a_{\sigma(n)}
\end{cases}$$

and the linear map

$$
\alpha_V
  \colon
\begin{cases}
  V^{(p)} \to\otimes^p V
  \\
  a\otimes \lambda\mapsto\lambda(a\otimes\cdots\otimes a)
\end{cases}$$

then the map $V^{(p)}\stackrel{\alpha_V}{\to}TS^p V\to TS^p V/s(\otimes^p V)$ is [[bijective]] and we define

$$
  \lambda_V
   \;\colon\;
  TS^p V\to V^{(p)}
$$

by

$$\lambda_V\circ s=0$$

and

$$\lambda_V \circ \alpha_V= id$$

If $A$ is a $k$-ring we have that $TS^p A$ is a $k$-ring and $\lambda_A$ is a $k$-ring morphism.

If $X=Sp_k A$ is the [[spectrum of a commutative ring]] we abbreviate $S^p X 0 S^p_k X \coloneqq Sp_k (TS^p A)$ and the following diagram is commutative.

$$\array{
X
&\stackrel{F_X}{\to}&
X^{(p)}
\\
\downarrow&&\downarrow
\\
X^p
&\stackrel{can}{\to}& S^p X
}$$

### For $E_\infty$-Rings
 {#ForEInfinityRings}

+-- {: .num_defn #EInfinityFrobenius}
###### Definition

Let $E$ be a [[E-infinity ring]] and $p$ a [[prime number]].
Then the _Frobenius morphism_ on $R$ is the composite morphism of [[spectra]]

$$
  R
    \overset{\Delta_p}{\longrightarrow}
  (R \wedge \cdots \wedge R)^{t C_p}
    \overset{prod^{t C_p}}{\longrightarrow}
  R^{t C_p}
$$

where

1. $C_p = \mathbb{Z}/p\mathbb{Z}$ denotes the [[cyclic group]] of order $p$;

1. $(-)^{t C_p}$ denotes the [[Tate spectrum]] of a spectrum with $C_p$-[[action]];

1. the [[smash product of spectra]] $R \wedge \cdots \wedge R$ is regarded with the $C_p$-action given by [[permutation]] of smash factors;

1. $\Delta_p$ dentotes the _[[Tate diagonal]]_ map

1. $prod^{t C_p}$ is the image of the $p$-fold product operation of the ring spectrum $prod \;\colon\; R \wedge \cdots \wedge R \to R$ under the [[(infinity,1)-functor]] which forms Tate spectra.

=--

([Nikolaus-Scholze 17, def. IV.1.1](#NikolausScholze17))

+-- {: .num_example #EInfinityFrobeniusReducesToOrdinaryFrobenius}
###### Example

Let $A \in CRing$ be an ordinary [[commutative ring]] and write $H A$ for its [[Eilenberg-MacLane spectrum]]. Then for $p$ a [[prime number]] the Frobenius homomorphism from def. \ref{EInfinityFrobenius}

$$
  H A \longrightarrow H A^{t C_p}
$$

coincides on the 0th [[stable homotopy group]] with the ordinary Frobenius homomorphism

$$
  A \longrightarrow A/p
$$

=--

([Nikolaus-Scholze 17, example IV.1.1](#NikolausScholze17))

## Properties

### For rings

* The Frobenius homomorphism is a [[natural transformation]] $Frob_p\colon Id_{Ring_p}\Rightarrow Id_{Ring_p}$ from the [[identity functor]] on the [[full subcategory]] of [[Ring]] containing all rings with [[characteristic]] $p$ to itself.

### For fields

* The Frobenius morphism on algebras is always [[injective]]. Note that the Frobenius morphism of schemes (see below) is *not* always a monomorphism.

* The [[image]] of the Frobenius morphism is the set of elements of $k$ with a $p$-th root and is sometimes denoted $k^{1/p}$.

* The Frobenius morphism is [[surjective]] if and only if $k$ is [[perfect field|perfect]].

### As elements of the Galois group
 {#AsElementsOfGaloisGroup}

Some powers of the Frobenius morphism canonically induce elements in the [[Galois group]] (...)

Review of the standard story is for instance in ([Snyder 02, section 1.5](#Snyder02)).
Further developments include  ([Dokchitser-Dokchitser 10](#DokchitserDokchitser10))

If $\mathbb{F}_{p^n}$ is a [[finite field]],then $Gal(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is [[generators and relations|generated]] by the Frobenius map $x\mapsto x^p$ (e.g. [Snyder 02, lemma 1.5.10](#Snyder02)).

(..)

See also at _[[Artin L-function]]_.

### For schemes

For the purposes below $k$ will be a [[perfect field]] of [[characteristic]] $p$>$0$.

* $X$ is [[smooth scheme|smooth]] over $k$ if and only if $F$ is a [[vector bundle]], i.e. $F_*\mathcal{O}_X$ is a free $\mathcal{O}_X$-module of rank $p$. One can study singularities of $X$ by studying properties of $F_*\mathcal{O}_X$.

* If $X$ is [[smooth scheme|smooth]] and [[proper scheme| proper]] over $k$, the sequence $0\to \mathcal{O}_X\stackrel{F^{ab}}{\to} F_*\mathcal{O}_X \to d\mathcal{O}_X\to 0$ is exact and if it [[split exact sequence|splits]] then $X$ has a lifting to $W_2(k)$.

+-- {: .num_prop}
###### Proposition
Let $X$ be a $k$-formal scheme (resp. a [[locally algebraic scheme]]) then $X$ is [[etale scheme|étale]] iff the [[Frobenius morphism]] $F_X:X\to X^{(p)}$is a monomorphism (resp. an isomorphism).
=--

The Frobenius as a morphism (natural transformation) of (affine) group schemes is one operation among other (related) operations of interest:

+-- {: .num_remark}
###### Remark
For any commutative affine [[group scheme]] $G$ the Frobenius- and the [[Verschiebung morphism]] correspond by "completed [[Cartier duality|Cartier duality]]''; i.e. we have

$$\hat D(V_G)=F_{\hat D(G)}$$
=--

For a more detailed account of the relationship of Frobenius-, [[Verschiebung morphism|Verschiebung-]] and [[homothety morphism]] see [Hazewinkel](#Hazewinkel)

## Related concepts

* [[Artin-Schreier sequence]]

* [[Weil conjecture]]

* [[restricted Lie algebra]]

* [[shtuka]]

* [[cyclotomic spectrum]]

## References

* {#Frobenius68} [[Ferdinand Georg Frobenius]], Vol 2, around p. 719 of _Gesammelte Abhandlungen_, Springer-Verlag, Berlin, 1968.

Lecture notes include

* [[Günter Tamme]], section II 4.2 of _[[Introduction to Étale Cohomology]]_

* [[James Milne]], section 27 of _[[Lectures on Étale Cohomology]]_

Further discussion of the relation to the [[Galois group]] includes

* {#Snyder02} [[Noah Snyder]], section 1.5 of _Artin L-Functions: A Historical Approach_, 2002 ([pdf](http://www.math.columbia.edu/~nsnyder/thesismain.pdf))

* {#DokchitserDokchitser10} Tim Dokchitser, Vladimir Dokchitser, _Identifying Frobenius elements in Galois groups_ ([arXiv:1009.5388](http://arxiv.org/abs/1009.5388))

See also

* Michel Demazure, _[[lectures on p-divisible groups]]_ [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

* [[Michiel Hazewinkel]], witt vectors. part 1, [arXiv:0804.3888v1](http://arxiv.org/abs/0804.3888v1){#Hazewinkel}

* Karen Smith, *Brief Guide to Some of the Literature on F-singularities*, American Institute of Mathematics

Discussion in the context of [[power operations]] on [[E-infinity rings]] is in

* {#Lurie} [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

Discussion for [[E-infinity rings]] via [[Tate spectra]] is due to

* {#NikolausScholze17} [[Thomas Nikolaus]], [[Peter Scholze]], _On topological cyclic homology_ ([arXiv:1707.01799](https://arxiv.org/abs/1707.01799))

A brief discussion in the context of algebraic geometry over rigs is in

* [[William Lawvere]], _Core Varieties, Extensivity, and Rig Geometry_ , TAC **20** no.14 (2008) pp.497-503. ([abstract](http://www.tac.mta.ca/tac/volumes/20/14/20-14abs.html))

[[!redirects Frobenius morphisms]]

[[!redirects Frobenius homomorphism]]
[[!redirects Frobenius homomorphisms]]

[[!redirects Frobenius endomorphism]]
[[!redirects Frobenius endomorphisms]]

[[!redirects Frobenius automorphism]]
[[!redirects Frobenius automorphisms]]

[[!redirects Frobenius map]]
[[!redirects Frobenius maps]]