# Exterior powers in tensor categories

Let $\mathcal{C}$ be a cocomplete $R$-linear tensor category. Many notions of commutative algebra can be internalized to $\mathcal{C}$. For example a commutative algebra is an object $A$ in $\mathcal{C}$ together with morphisms $e : 1 \to A$ (unit) and $m: A \otimes A \to A$ (multiplication) satisfying the usual laws. The $n$-th symmetric power $\text{Sym}^n(X)$ of an object $X$ is the quotient of $X^{\otimes n}$ by identifying $x_1 \otimes ... \otimes x_n = x_{\sigma(1)} \otimes ... \otimes x_{\sigma(n)}$, so formally it is defined as a coequalizer of the $n!$ symmetries $X^{\otimes n} \to X^{\otimes n}$. Then $\text{Sym}(X) = \bigoplus_{n\geq 0} \text{Sym}^n(X)$ is a commutative algebra and in fact $\text{Sym}$ is left adjoint to the forgetful functor $\mathsf{CAlg}(\mathcal{C}) \to \mathcal{C}$.

But now what about the exterior power $\Lambda^n(X)$? It is clear how to define $X^{\otimes n}$ modulo $x_1 \otimes ... \otimes x_n = \text{sgn}(\sigma) \cdot x_{\sigma(1)} \otimes ... \otimes x_{\sigma(n)}$ in this context, which one might call the anti-symmetric power $\mathrm{ASym}^n(X)$. The correct definition of the exterior power also has to mod out $$... \otimes a \otimes ... \otimes a \otimes ... = 0,$$ for example because we want to have that $\Lambda^p(1^{\oplus n}) \cong 1^{\oplus \binom{n}{p}}$. But I have no idea how to internalize this to $\mathcal{C}$, even for $n=2$. The reason is that there is no morphism $X \to X \otimes X$ which acts like $a \mapsto a \otimes a$. Another idea would be to define $\Lambda(X)$ as a graded-commutative algebra object with the usual universal property, classifying morphisms $f$ on $X$ which satisfy something like $f(x)^2=0$, but again it is unclear how to formulate this in $\mathcal{C}$.

If this is not possible at all, which additional structure on $\mathcal{C}$ do we need in order to define exterior powers within them? Is this some categorified $\lambda$-ring structure? This structure should be there in the case of usual module categories (over rings or even ringed spaces). Of course there is no problem when $2 \in R^*$, because then the exterior power equals the anti-symmetric power. The question was also discussed in a blog post.

Here is a more specific (and a bit stronger) formulation: Is there some $R[\Sigma_n]$-module $T$, such that for every $R$-module $M$, we have that $T \otimes_{R[\Sigma_n]} M^{\otimes n} \cong \Lambda^n M := M^{\otimes n}/(... \otimes x ... \otimes x ...)$? Because then we could define $\Lambda^n X := T \otimes_{R[\Sigma_n]} X^{\otimes n}$ for $X \in \mathcal{C}$.

Concerning the "hidden extra structure" in the case of modules: Let the base ring be $\mathbb{Z}$, or more generally a commutative ring $R$ in which $r^2 - r \in 2R$ for all $r \in R$; this includes boolean rings such as $\mathbb{F}_2$ and also $\mathbb{Z}/n$. If $M$ is an $R$-module, then there is a well-defined(!) homomorphism $M^{\otimes~ n-1} \to \text{ASym}^n(M), x_1 \otimes ... \otimes x_n \mapsto x_1 \wedge x_1 \wedge ... \wedge x_n$, and its cokernel is $\Lambda^n(M)$.

Concerning non-linear tensor categories, I've asked here a similar question.

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Why do you believe "the correct one has to mod out be $\dots \otimes a\otimes \dots \otimes a \otimes \dots = 0$"? As you say, such a map does not exist in the categorical context, and it's not at all clear to me that it's the natural condition. For example, in the super/graded world, it can be very useful to play off the fact that $\wedge^n(X) = \pi^{-n}\operatorname{Sym}^n(\pi X)$, where $\pi$ is the "shift" functor. This fails when $X$ has odd part if you try to use a "mod out by squares" definition. –  Theo Johnson-Freyd Jul 18 '11 at 11:46
Agree. I mean with the wrong definition we have $\wedge^n A = A/2A$ for $n \geq 2$, but it should be $0$. –  Martin Brandenburg Jul 18 '11 at 13:14
There is no "right" and "wrong" definition. There simply exist two different mathematical objects, that happen to agree when 2 is invertible. One of them happens to produce 2-torsion in some cases. –  André Henriques Jul 18 '11 at 15:34
@Martin: What is the simplest example of a category in which you'd like to interpret ᐽ(X), and where you don't know how to do it? –  André Henriques Jul 18 '11 at 15:43
@Theo: Surely there are some super settings where the "right" notion of $Sym^n$ (for the purpose at hand) involves the square of an odd-degree element being zero. The $\Lambda^n$ that is related to this by your equation is what Martin is calling the exterior power (in which the square of an even-degree element is zero). Likewise there are some settings where what you are calling $Sym^n$ is the "right" thing; and for these the $\Lambda^n$ that is given by your equation is what Martin is calling the skew-symmetric power. –  Tom Goodwillie Jul 19 '11 at 17:05

Deligne (Categories Tannakiennes, 1990, p165) defines it to be the image of the antisymmetrisation $a=\sum(-1)^{\epsilon(\sigma)}\sigma\colon X^{\otimes n}\rightarrow X^{\otimes n}$.
No. This map from $X^{\otimes n}$ to itself kills tensors of the form $\dots \otimes a\otimes\dots\otimes a\otimes\dots$, and at least for free modules its image coincides with $\Lambda^nX$. The expression "antisymmetrization map", like many other terms in this area, can be misleading, because one could also use it for the map from $X^{\otimes n}$ to the largest quotient on which the action (with signs) of the symmetric becomes trivial. –  Tom Goodwillie Jul 19 '11 at 16:51