The n-Category Café

The 2-category $$2\mathrm{Vect}$$ (in the present context) is that of categories internal to vector spaces.

The 2-category $$2\mathrm{Term}$$ is that of chain complexes of length 2.

Notice that such chain complexes are really nothing but morphisms $$V_1\stackrel{f}{\to }{V}_0$$ of vector spaces. So an object in $2\mathrm{Term}$ is nothing but a linear map.

Morphisms of 2-term chain complexes are nothing but commuting squares between such linear maps

$$\begin{array}{ccc}{V}_1& \stackrel{f}{\to }& V_0\\ \downarrow & & \downarrow \\ V{\prime }_1& \stackrel{f\prime }{\to }& V{\prime }_0\end{array}.$$

Therefore Loday and Pirashvili call this category not $2\mathrm{Term}$ but

$$LM$$

(“Linear Maps”).

They would however probably have thought of a different name than that had they thought of the next level categorification of their setup. That they didn’t is apparently largely due to the fact, as far as I can see at least, that one of main facts they are interested in in this paper is that a couple of ordinary non-Lie algebras, like bialgebras, Leibniz algebras, become Lie objects internal to $2\mathrm{Term}$.

We will of course rephrase these statements in the form

Every Leibniz algebra $g$ gives rise to a strict Lie 2-algebra $$g\to g_{\mathrm{Lie}}.$$

So, recall first the standard fact described in HDA VI:

There is a (slightly non-canonical) equivalence $$2\mathrm{Vect}\simeq 2\mathrm{Term}.$$

Given a 2-vector space $V$, we define the corresponding 2-term chain complex to be $$\ker (s)\stackrel{t}{\to }\mathrm{Obj}(V).$$

The 2-category $2\mathrm{Vect}$ has an obvious monoidal structure obtained by tensoring internal to $\mathrm{Vect}$.

Let $V_1\otimes V_2$ be the tensor product of two 2-vector spaces. Then, since $${\ker }_{s_{V_1\otimes V_2}}=\langle \begin{array}{c}0\\ {\downarrow }^f\\ t(f)\end{array}\otimes {\mathrm{Id}}_{a_2}\oplus {\mathrm{Id}}_{a_1}\otimes \begin{array}{c}0\\ {\downarrow }^{f\prime }\\ t(f\prime )\end{array}\rangle$$ we find that $V_1\otimes V_2$ corresponds to the 2-term chain complex given by $$\ker (s_1)\otimes \mathrm{Obj}(V_2)\oplus \mathrm{Obj}(V_2)\otimes \ker (s_2)\stackrel{t_1\otimes \mathrm{Id}\oplus \mathrm{Id}\otimes t_2}{\to }\mathrm{Obj}(V_1)\otimes \mathrm{Obj}(V_2).$$

Equipped with this non-obvious tensor product

$$\left(\begin{array}{c}V\\ \downarrow \\ W\end{array}\right)\otimes \left(\begin{array}{c}V\prime \\ \downarrow \\ W\prime \end{array}\right):=\left(\begin{array}{c}(V\otimes W\prime )\oplus (W\otimes V\prime )\\ \downarrow \\ W\otimes W\prime \end{array}\right)$$

$2\mathrm{Term}$ becomes a symmetric monoidal closed category. The equivalence $2\mathrm{Vect}\simeq 2\mathrm{Term}$ extends to an equivalence of symmetric monoidal categories.

The internal Hom-object is define din the very last lines of Loday&Pirashvili’s paper.

By the equivalence, this then also determines the internal hom in Baez-Crans 2-vector spaces $$2\mathrm{Vect}.$$

Loday and Pirashvili have a couple of further interesting observations concerning various kinds of 2-algebras as algebras internal to Baez-Crans 2-vector spaces.