## Path-linearizable categories

### Marta Bunge – 5 September 2001

A category B is said to be path-linearizable [BN, BF] if it satisfies an ``interval glueing condition''. The latter (IG) is seen to be equivalent to the conjunction of two independent properties, namely (C) ``cancellation'' and (FI) ``factorization lifting''. Examples of path-linearizable categories are: any category free on a directed graph (in particular, the additive monoid N of natural numbers) and the additivive monoid R+ of non-negative real numbers. It is proved in [BN] (see also [BF]) that over a path-linearizable category B, the category UFL/B of unique factorization lifting functors over B is a Grothendieck topos.
The history of the above mentioned Bunge-Niefield theorem [BN] (Theorem 4.5) is a comedy of errors. It had been claimed (with a sketchy proof) by F. Lamarche [Lam] that for any small category B, the category UFL/B of unique factorization lifting functors (or of discrete Conduché fibrations [C] but also [G]) is (always) a topos. This claim remained undisputed (at best, ignored) for several years. In a preliminary version to [BN], submitted to P. T. Johnstone (as editor) for the Journal of Pure and Applied Algebra, Bunge and Niefield showed that the Lamarche claim was equivalent to the existence of a right adjoint to the canonical inclusion of UFL/B-->Cat/B thus making UFL/B into a ``model-generated category'' [N] and hence (in this case, also) a topos, which we then proceeded to ``prove''. The statement, in this form, was shown by the said editor to be false, by exhibiting a counterexample (the independence square to the fact that UFL/B is in general a coreflective subcategory of Cat/B. This led us to immediately locate the gap in our proof and to introduce accordingly the condition (IG) to repair it. It also led the editor to publish his remarks in [J] even before our paper [BN] could appear in print. An alternate proof, examining the role of each of the two conditions (C) and (FI) was then given in [BF], profitting this time partly from some useful remarks made in [J].
In this talk I intend to add some new remarks to what is contained in [BN] and [BF], to wit:
Establish the validity of an ``ambush/UFL'' factorization in the context of path-linearizable categories, based on comprehension for a suitable fibration with $\Sigma$ and a terminal T, Relate the path-linearizable categories to the Möbius categories of [CLL] and give a combinatorial characterization of UFL functors in terms of homomorphisms of incidence algebras, and Discuss the notion of category of processes that arises from assuming the existence of a ``duration'' (or length, or shape) functor on the control (or configuration) category, in connection with [Law], Pose some questions regarding the 2-dimensional analogues of the UFL functors as defined in [S], and also regarding the version of the UFL functors for geometric morphisms introduced in [BN].
Discussions with Ross Street and Bill Lawvere are gratefully acknowledged.
Bibliography:
[BF]M. Bunge and M. Fiore, Unique factorization lifting functors and categories of linearly-controlled processes, Math. Str. in Comp. Sci. 10 (2000) 137-163.
[BN] M. Bunge and S.B. Niefield, Exponentiability and single universes, J.Pure Appl. Algebra 148-3 (2000) 217-250.
[C] F. Conduché, Au sujet de l'existence d'adjoints à droîte aux foncteurs ``image reciproque'' dans la catégorie des cetégories, C.R.Acad. Sci. Paris 275 (1972) A891-894.
[CLL] M. Content, F. Lemay and P. Leroux, Catégories de Möbus et fonctorialités: un cadre général pour l'inversion de Möbius, J. Combinatorial Theory Series A 28 (1980) 169-190.
[FK] P. Freyd and G.M. Kelly, Categories of continuous functors I, J. Pure Appl. Alg 2 (1972) 169-191. Erratum: J. Pure Appl. Alg 4 (1974) 121.
[G] J. Giraud, Méthode de la descente, Bull. Math. Soc. Memoire 2, 1964.
[J] P.T. Johnstone, A note of discrete Conduché fibrations, Theory and Application of categories 5 (1999) 1-11.
[Lam] F. Lamarche, On a new class of toposes, Unpublished lecture, PSSL Utrecht (1996).
[Law] F.W. Lawvere, State categories and response functors, Unpublished manuscript (1986).
[Mac] S. MacLane, Categories for the Working Mathematician, Springer-Verlag, 1971.
[N] S.B. Niefield, Cartesianess, Ph.D. thesis, Rutgers University, 1978.
[S] R. Street, Categorical structures, in : Handbook of Algebra 1, North-Holland, 1998, 529-577.
[SW] R. Street and R.F.C. Walters, The comprehensive factorization of a functor, Bull. Amer. Math. Soc. 79 (1973) 936-941.

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