nLab
generalized kernel

Contents

Idea

In an Ab-enriched category, it is natural to produce an image factorization of a morphism by first forming its kernel and then the cokernel of the kernel. Similarly, in a regular category we can produce an image factorization by first forming the kernel pair and then the coequalizer of the kernel pair. Several similar situations arise in the study of 2-categories as well. The theory of generalized kernels in enriched categories subsumes all of these examples.

Generalized kernels and their quotients

Let VV be a cosmos and let 2\mathbf{2} denote the interval category regarded as a VV-category, i.e. it has two objects 00 and 11, with hom(0,0)=hom(1,1)=hom(0,1)=Ihom(0,0) = hom(1,1) = hom(0,1) = I (the unit object of VV) and hom(1,0)=hom(1,0) = \emptyset (the initial object of VV). Let HH be a VV-category which contains 2\mathbf{2} as a full subcategory, and let JJ be the full subcategory of HH containing all the objects except 11. Then the inclusions of JJ and 2\mathbf{2} into HH induce a profunctor K:2JK\colon \mathbf{2} ⇸ J. This is the input data for a notion of kernels.

Now suppose that CC is a VV-category with sufficiently many limits and colimits, and that f:xyf\colon x\to y is a morphism in CC. Then ff can be identified with a VV-functor ulcornerfurcorner2C\ulcorner f \urcorner \mathbf{2}\to C. The kernel of ff is defined to be the KK-weighted limit of ulcornerfurcorner\ulcorner f \urcorner , which is a VV-functor ker(f):JCker(f)\colon J\to C. By construction of KK, it can equivalently be described as the (enriched, pointwise) right Kan extension of ulcornerfurcorner\ulcorner f \urcorner from 2\mathbf{2} into HH, followed by restriction to JJ. Since the inclusion of 2\mathbf{2} into HH is fully faithful, it follows that ker(f)(0)=ulcornerfurcorner(0)=xker(f)(0) = \ulcorner f \urcorner (0) = x.

Similarly, let MM be a VV-functor JCJ\to C; we call such a functor kernel data. The quotient of MM defined to be its KK-weighted colimit. This is a functor 2C\mathbf{2}\to C, i.e. a single morphism in CC. Again, by construction of KK, the quotient of MM can equivalently be described by the left Kan extension of MM to HH, followed by restriction to 2\mathbf{2}, and since JHJ\to H is fully faithful, the source of the quotient of MM is the object M(0)M(0).

Since colimits and limits weighted by a fixed profunctor are adjoint to each other (or equivalently, since left and right Kan extension are left and right adjoint to restriction), we obtain an adjunction

quot:[J,C][2,C]:ker quot: [J,C] \;\rightleftarrows\; [\mathbf{2},C] :ker

Moreover, by the remarks above, for any object xCx\in C this adjunction restricts to an adjunction

quot:[J,C] xx/C:ker quot: [J,C]_x \;\rightleftarrows\; x/C :ker

where [J,C] x[J,C]_x denotes the subcategory of [J,C][J,C] on those functors MM such that M(0)=xM(0)=x. Of course the analogous subcategory [2,C] x[\mathbf{2},C]_x is just the coslice category x/Cx/C.

Of particular interest is the counit of this adjunction at a morphism f:xyf\colon x\to y, which is a morphism quot(ker(f))fquot(ker(f)) \to f in x/Cx/C. In other words, it is a factorization of ff. Often the factorizations produced in this way are familiar.

Examples

  1. With V=SetV=Set, let HH be the category

    z01 z \;\rightrightarrows\; 0 \to 1

    with the two composites z1z\to 1 being equal. That is, HH is the “walking fork.” Then kernel data is a pair of parallel arrows, the kernel of a morphism is its kernel pair, and the quotient of a parallel pair is their coequalizer. In a regular category, the factorization produced in this way is the usual image factorization.

    Note that since coequalizers are epic, the full image of quot:[J,C][2,C]quot\colon [J,C] \to [\mathbf{2},C] is a preorder, and thus in this case the adjunction is idempotent. This yields the familiar fact that a morphism is a coequalizer iff it is the coequalizer of its kernel pair, and a parallel pair is a kernel pair iff it is the kernel pair of its coequalizer.

  2. Again with V=SetV=Set, let HH be the category

    0 z 1 \array{ 0 \\ & \searrow \\ z & \to & 1 }

    Then kernel data is just a pair of objects, the kernel of a morphism f:xyf\colon x\to y is the pair (y,x)(y,x), and the quotient of a pair (y,x)(y,x) is the coproduct injection x(xy)x \to (x\sqcup y). The factorization of f:xyf\colon x\to y obtained in this way is x(xy)[f,id]yx \to (x\sqcup y) \overset{[f,id]}{\to} y.

  3. With V=AbV=Ab, let HH be the Ab-enriched category with three objects z,0,1z,0,1 and H(z,z)=H(0,0)=H(1,1)=H(z,0)=H(0,1)=H(z,z) = H(0,0) = H(1,1) = H(z,0) = H(0,1) = \mathbb{Z} and all other hom-groups being 00. Then a diagram of shape HH in an Ab-category consists of a pair of composable arrows whose composite is zero. Then kernel data is also just a single morphism, the kernel of a morphism is its kernel in the usual sense, and the quotient of a morphism (considered as kernel data) is its cokernel in the usual sense. The resulting factorization is again the usual image factorization, and the adjunction is again idempotent.

  4. With V=CatV=Cat, let HH be the (strict) 2-category with three objects z,0,1z,0,1 and H(0,1)=*H(0,1) = *, H(z,1)=2H(z,1) = \mathbf{2}, and H(z,0)H(z,0) the “walking parallel pair()(\cdot \rightrightarrows \cdot). Then kernel data in a 2-category consists of a pair of parallel 2-cells, and the quotient of such a pair is their coequifier. In CatCat at least, the kernel f:xyf\colon x\to y of a functor is the category of parallel pairs in xx which become equal in yy, and the resulting factorization of ff consists of an bijective-on-objects-and-full functor followed by a faithful one. Since bijective-on-objects-and-full functors are epic in CatCat, the adjunction is again idempotent.

  5. Again with V=CatV=Cat, let HH be the 2-category with three objects y,0,1y,0,1 and H(0,1)=*H(0,1)=*, H(y,0)H(y,0) the discrete category with two objects, and H(y,1)=2H(y,1)=\mathbf{2}. Then kernel data is a pair of parallel 1-morphisms, and the quotient of such a pair is their coinserter. The kernel of a morphism f:xyf\colon x\to y is the comma object (f/f)(f/f).

  6. Once again with V=CatV=Cat, let HH be the 2-category

    0 z 1 \array{ && 0\\ & \nearrow & \cong & \searrow \\ z && \to && 1}

    Then kernel data is again a single arrow, its quotient is its pseudocolimit, and the kernel of a morphism is its pseudolimit. The resulting factorization is one of the ones arising in the canonical model structure on any 2-category.

  7. Continuing with V=CatV=Cat, let HH be the 2-category on z,0,1z,0,1 with H(0,1)=*H(0,1)=*, H(z,0)=2H(z,0) = \mathbf{2}, and H(z,1)H(z,1) the walking isomorphism. Then kernel data is a 2-cell, and its quotient is its coinverter. The kernel of a morphism is sometimes called its invertee?. Since coinverters are epic, the adjunction is idempotent: thus a morphism is a coinverter iff it is the coinverter of its invertee, and a 2-cell is an invertee iff it is the invertee of its coinverter.

  8. Finally with V=CatV=Cat, let HH be the 3-truncated augmented simplex category with additional 2-cells added, in such a way that kernel data is codescent data and the quotient of such data is a codescent object?. Then the factorization produced in CatCat consists of a bijective-on-objects functor followed by a fully faithful one.

Factorization systems

In some of the above examples, the resulting factorization of a morphism is not what one would hope for. In particular, it need not give a factorization system of any kind. However, the adjunction does induce a comonad Q 1=quotkerQ_1 = quot \circ ker on [2,C][\mathbf{2},C] such that dom(Q 1(f))=dom(f)dom(Q_1(f)) = dom(f). This is the “left half” of the factorization obtained in this way; the “right half” R 1R_1 is merely a pointed endofunctor such that cod(R 1(f))=cod(f)cod(R_1(f)) = cod(f). The algebras for R 1R_1 are usually what one would hope for as the “right class” of the (weak) factorization systems intuitively associated to the above examples.

Now this data is exactly the required input for the construction of a free algebraic weak factorization system. If CC is sufficiently well-behaved (such as locally presentable), then this free nwfs exists, and gives the factorizations we would hope for in most cases. It is constructed by (possibly transfinite) iteration, i.e. we take the quotient of the kernel, then take the quotient of the kernel of the right half of the resulting factorization, and so on. The resulting monad RR is usually algebraically-free on R 1R_1, and thus the RR-algebras are again just what we would expect.

For example, in the coinverter-invertee example above, the right clas of the factorization system produced in this way consists of conservative functors.

References

This sort of idea has been thought of by various people at various times. The above version, especially the view of factorization systems, is a slight modification of one presented by Richard Garner to the Sydney category seminar in February 2010. Other references include

  • Betti, Schumacher, and Street, “Factorizations in bicategories” (unpublished)

  • Betti, “Adjointness in descent theory”, JPAA

  • Mathieu Dupont, Abelian categories in dimension 2, arXiv:0809.1760, section 1.2.2

Revised on December 7, 2011 04:05:22 by Mike Shulman (71.136.231.40)