symmetric monoidal (∞,1)-category of spectra
A Frobenius algebra in a monoidal category is a quintuple $(A, \delta, \epsilon, \mu, \eta)$ such that
In terms of string diagrams, this definition says:
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.
Frobenius algebras were originally formulated in the category Vect of vector spaces with the following equivalent definition:
A Frobenius algebra is a unital, associative algebra $(A, \mu, \eta)$ equipped with a linear form $\epsilon : A \rightarrow k$ such that $\epsilon\mu$ is a non-degenerate pairing. I.e. the induced map
is an isomorphism of $V$ with its 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.
There are about a dozen equivalent definitions of a Frobenius algebra. Ross Street (2004) lists most of them.
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.
We can define ‘commutative’ or ‘symmetric’ Frobenius algebras in any symmetric monoidal category. A Frobenius algebra $A$ is symmetric if
where $S_{A,A} : A \otimes A \to A \otimes A$ is the symmetry. 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 and Nakayama says that in the category 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, meaning that for any field $K$ extending $k$, $A \otimes_k K$ is a semisimple algebra over $K$.
If $\mu \circ \delta = 1$, a Frobenius algebra is said to be special. In the category of vector spaces, any element $a$ of an associative unital algebra gives a left multiplication map
which in turn gives a bilinear pairing $g: A \times A \to k$ defined by
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
In this case, there is just one way to make $A$ into a special Frobenius algebra, namely by taking the counit to be
(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).
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 $1 \in G$:
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:
We should warn the reader that Rosebrugh et al (2005) 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.
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 theory.
A Frobenius algebra $A$ in a monoidal category is an object dual to itself.
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 usual triangular equations hold. The maps are defined by
and one of the triangular equations uses one of the Frobenius laws and unit and counit axioms to derive the following commutative diagram:
The other triangular equation uses the other Frobenius law and unit and counit axioms.
As a result, we see that in the monoidal category $Mod_k$ of modules over a commutative ring $k$, Frobenius algebras $A$ considered as modules over $k$ are finitely generated and projective. This is because $A \otimes_k -$, being adjoint to itself, is a left adjoint and 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.
Certain kinds of Frobenius algebras have nice PROPs or PROs. The PRO for Frobenius algebras is the monoidal category of planar thick tangles, as noted by Aaron Lauda Lauda (2006) and illustrated here:
Lauda and Pfeiffer Lauda (2008) showed that the PROP for symmetric Frobenius algebras is the category of ‘topological open strings’, since it obeys this extra axiom:
The PROP for commutative Frobenius algebras is 2Cob?, as noted by many people and formally proved by Lowell Abrams Abrams (1996). This means that any commutative Frobenius algebra gives a 2d TQFT. See Kock (2006) for a history of this subject and Kock (2004) for a detailed introduction. In 2Cob, the circle is a Frobenius algebra. The monoid laws look like this:
The comonoid laws look like this:
The Frobenius laws look like this:
and the commutative law looks like this:
The PROP for special commutative Frobenius algebras is Span(FinSet), as proved by Rosebrugh, Sabadini and Walters. This is worth comparing to the PROP for commutative bialgebras, which is Cospan(FinSet). For details, see Rosebrugh et al (2005), and also Lack (2004).
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
Frobenius algebras were introduced by Brauer and Nesbitt and were named after Ferdinand Frobenius.
See for instance
Their role in 2-dimensional TQFT is discussed for instance in
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)
Richard Garner, Three investigations into linear logic , PhD report Cambridge 2006. (pdf)
D. Kartsaklis, M. Sadrzadeh, S. Pulman, B. Coecke, Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras , arXiv:1401.5980 (2014). (pdf)
See also
B. Fauser, Some Graphical Aspects of Frobenius Structures , preprint (2012). arXiv:1202.6380
Stephen Lack (2004), Composing PROPs, Theory and Applications of Categories 13, 147–163. (web)
R. F. C. Walters, R. J. Wood, Frobenius Objects in Cartesian Bicategories , TAC 20 no. 3 (2008) pp.25-47. (pdf)
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)
(TO ADD: stuff from this MO post) (TO DO: $\epsilon$ is used for the comultiplication in the beginning, then for the linear form, and in the definition of symmetry for the bilinear pairing on a Frobenius algebra. Use three different letters for these)