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

$$\array{ V_1 &\stackrel{f}{\to}& V_0 \\ \downarrow &&\downarrow \\ V'_1 &\stackrel{f'}{\to}& V'_0 } \,.$$

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

$$L M$$

(“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 $$\mathrm{ker}(s) \stackrel{t}{\to} \mathrm{Obj}(V) \,.$$

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 $$\mathrm{ker}_{s_{V_1 \otimes V_2}} = \left\langle \array{ 0 \\ \downarrow^f \\ t(f) } \otimes \mathrm{Id}_{a_2} \;\oplus\; \mathrm{Id}_{a_1} \otimes \array{ 0 \\ \downarrow^{f'} \\ t(f') } \right\rangle$$ we find that $V_1 \otimes V_2$ corresponds to the 2-term chain complex given by $$\mathrm{ker}(s_1)\otimes \mathrm{Obj}(V_2) \oplus \mathrm{Obj}(V_2) \otimes \mathrm{ker}(s_2) \stackrel{t_1\otimes \mathrm{Id}\oplus \mathrm{Id}\otimes t_2}{\to} \mathrm{Obj}(V_1) \otimes \mathrm{Obj}(V_2) \,.$$

Equipped with this non-obvious tensor product

$$\left( \array{ V \\ \downarrow \\ W } \right) \otimes \left( \array{ V' \\ \downarrow \\ W' } \right) := \left( \array{ (V \otimes W' ) \oplus (W \otimes V') \\ \downarrow \\ W \otimes W' } \right)$$

$2\mathrm{Term}$ 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.