nLab category of factorizations

History

In his monograph on (co-)extensions of monoids, Leech introduced a construction that had also been used for other purposes by MacLane, see twisted arrow category. A short time later Charles Wells distributed a preprint which made the short trip from monoids to small categories with a fixed set of objects. This introduced the constructions below but then the subject lay fallow until work by Baues and Wirshing introduced what is now usually called Baues-Wirsching cohomology. Compare also factorization category.

Definition

The following constructions were used by Baues and Wirsching and we will more or less adapt their terminology and notation.

Given a small category $I$, one defines a category of factorizations $FI$ as follows. The objects of $FI$ are the morphisms of $I$. A morphism from $x:i\to j$ to $y : k\to l$ is a commutative square of the form (note the direction of arrows!)

$$\array{ i & \stackrel{v}\longleftarrow & k\\ {x}^{\mathllap{}}\downarrow&&\downarrow{y}^{\mathrlap{}}\\ j &\stackrel{u}\longrightarrow & l }$$

In other words, it is a pair $(u,v)$ which factorizes $y = u\circ x \circ v$. The composition is defined in the obvious way: $(u',v')\circ (u,v) := (u'\circ u, v\circ v')$. For a morphism $f$ in $I$, one usually denotes $D(f)$ by $D_f$ and uses the abbreviation $D_i = D_{id_i}$ for every object $i$ in $I$. Other conventions include $u_* = D(u,Id) : D_x\to D_{ux}$ and $v^* = D(Id, v) : D_x\to D_{xv}$.

Natural systems

A natural system on $I$ is a functor $D : FI\to Ab$.

The evident notion of a non-Abelian natural system does not behave that well. A lax version (introduced by Wells) handles that case.

Simple examples of natural systems are provided from the consideration of the sequence of functors

$$FI \stackrel{(i\overset{x}\to j)\mapsto (i,j)}\longrightarrow I^{op}\times I \stackrel{(i,j)\mapsto j}\longrightarrow I$$

which implies that any functor $I\to Ab$ and, more generally, any functor $I^{op}\times I\to Ab$ provides (after precomposing with the canonical functor from $FI$) a natural system on $I$.

Natural systems on $I$ are the coefficients of a Baues-Wirsching cohomology of small categories. Namely, the Baues-Wirsching cochain complex $C^*(I,D)$ has

$$C^n(I,D) = \prod D_{x_1\cdots x_n}$$

where the product is over all $i_n\stackrel{x_n}\longrightarrow \ldots \stackrel{x_1}\longrightarrow i_0$.

References

(A more extensive list may be found at the entry on Baues-Wirsching cohomology.)

• H.-J. Baues, Algebraic Homotopy, Cambridge studies in advanced mathematics 15, Camb, Univ. Press 1989.

• H.-J. Baues, G. Wirsching, Cohomology of small categories, J. Pure Appl. Alg. 38 (1985), 187-211.

• T. Pirashvili, On the center and Baues-Wirsching cohomology, Georgian Math. J. 16 (2009) 1, 131-144

• C. Wells, Extension theories for categories (preliminary report) (1979), available here

  http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf).