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Loday-Pirashvili category

Overview

Given any category $C$, one can define the arrow category $Arr(C)$ of $C$, whose objects are morphisms in $C$ and whose morphisms are commutative squares. If $C$ is the category of vector spaces (or some other $k$-linear closed symmetric monoidal category with equalizers) one can define the infinitesimal or Loday–Pirashvili (LP) tensor product on the category of arrows, as well as an inner hom, equipping the category $\mathrm{Arr}C$ with a structure of a $k$-linear closed symmetric monoidal category.

The LP-tensor product is

$$(f:V_1\to V_0)\otimes (g:W_1\to W_0):=(V_0\otimes g+f\otimes W_0:V_0\otimes W_1\oplus V_1\otimes W_0\to V_0\otimes W_0).$$(f:V_1\to V_0)\otimes (g:W_1\to W_0):= (V_0\otimes g + f\otimes W_0: V_0\otimes W_1 \oplus V_1\otimes W_0\to V_0\otimes W_0).

This is a truncation of the tensor product of chain complexes where $V_1\otimes W_1$ is dropped.

The inner hom is rather interesting: $\mathrm{Hom}(f,g)=(p:{\mathrm{Hom}}_1(f,g)\to {\mathrm{Hom}}_0(f,g))$, where ${\mathrm{Hom}}_0(f,g)$ is the equalizer of two morphisms

$$\mathrm{hom}(V_0,W_0)\oplus \mathrm{hom}(V_1,W_1)\to \mathrm{hom}(V_1,W_0),$$\mathrm{hom}(V_0,W_0)\oplus\mathrm{hom}(V_1,W_1)\to\mathrm{hom}(V_1,W_0),

namely precomposing the first summand with $f$ and postcomposing the second summand with $g$ (where $\mathrm{hom}$ is the ordinary inner hom in $C$), and where ${\mathrm{Hom}}_1(f,g)$ is the equalizer of two morphisms

$${\mathrm{Hom}}_0(f,g)\oplus \mathrm{hom}(V_0,W_1)\to {\mathrm{Hom}}_0(f,g),$$\mathrm{Hom}_0(f,g)\oplus\mathrm{hom}(V_0,W_1)\to \mathrm{Hom}_0(f,g),

namely the identity and the map which replaces the lower component with the postcomposition by $g$ applied on $\mathrm{hom}(V_0,W_1)$ and keeps the upper component. Finally, $p$ is the natural projection.

In the case of vector spaces this means that we have diagonal lifts in squares such that the lower square commutes but not necessarily the upper, i.e. $\mathrm{Hom}(f,g)$ is the space consisting of all triples $(u_1,u_0,\varphi )$ where $u_1:V_1\to W_1$, $u_0:V_0\to W_0$ and $\varphi :V_0\to W_1$ such that $g\circ u_1=u_0\circ f$ and $u_0=g\circ \varphi$ while one does not require $\varphi \circ f=u_1$.

There are a number of remarkable functors relating internal algebras in LP, Lie algebras in LP etc., to or from some other categories of algebras. For example the categories of left Leibniz algebras and of right Leibniz algebras embed as full subcategories into the category of internal Lie algebras in LP. This embedding has an adjoint. Notice that because of truncation, being a Lie algebra in LP is a bit less than a (strict) $2$-Lie algebra (a requirement in degree $2$ is dropped).

Literature and discussions

• Jean-Louis Loday, Teimuraz Pirashvili, The tensor category of linear maps, Georg. Math. J. vol. 5, n.3 (1998) 263–276.

• nlab

Tim: Methinks that we need some comment on the evident connection with Baez–Crans 2-vector space. I think I remember seeing some paper on 2-vector spaces that mentions the connection. Whether or not it exploited that connection has slipped my memory. Can Zoran say something on this?

Zoran Surely in char 0, internal categories to vector spaces are the same as 2-term chain complexes, but if one translates strict associative algebra, Lie algebra etc. internal to the categories of internal categories in ${\mathrm{Vec}}_k$ then one has more on the internal category side then on LP side because of the truncation of the tensor product. So every strict Lie algebra in Baez-Crans 2-vector spaces gives an examples of an internal Lie algebra in LP but not other way around. Eventually I will put some treatment of this, but it is not that simple to write it clearly, so it will wait a bit for now.