nLab category of factorizations

Categories of factorizations


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.


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

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

i v k x y j u l\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)(u,v) which factorizes y=uxvy = u\circ x \circ v. The composition is defined in the obvious way: (u,v)(u,v):=(uu,vv)(u',v')\circ (u,v) := (u'\circ u, v\circ v'). For a morphism ff in II, one usually denotes D(f)D(f) by D fD_f and uses the abbreviation D i=D id iD_i = D_{id_i} for every object ii in II. Other conventions include u *=D(u,Id):D xD uxu_* = D(u,Id) : D_x\to D_{ux} and v *=D(Id,v):D xD xvv^* = D(Id, v) : D_x\to D_{xv}.

Natural systems

A natural system on II is a functor D:FIAbD : 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(ixj)(i,j)I op×I(i,j)jI 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 IAbI\to Ab and, more generally, any functor I op×IAbI^{op}\times I\to Ab provides (after precomposing with the canonical functor from FIFI) a natural system on II.

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

C n(I,D)=D x 1x n C^n(I,D) = \prod D_{x_1\cdots x_n}

where the product is over all i nx nx 1i 0i_n\stackrel{x_n}\longrightarrow \ldots \stackrel{x_1}\longrightarrow i_0.


(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

Last revised on December 6, 2011 at 05:44:41. See the history of this page for a list of all contributions to it.