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Descent on 2-Fibrations and Strongly 2-Regular 2-Categories

Profile image of Claudio HermidaClaudio Hermida

2000, Applied Categorical Structures

https://doi.org/10.1023/B:APCS.0000049311.17100.DA
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Abstract

We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated nerve allows the definition of the 2-category of descent data for q. When the 2-fibration admits direct images, we provide the analogous of the Beck-Bénabou-Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q * Σq. We introduce a notion of 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations cod : Fib(R) → R admits direct images. In this context, we show that essentiallysurjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.

Figures (11)
where the the 2-cell \ corresponds to a comma-object, the morphism q is a fibration iff yy: q— q' has a right-adjoint in K/C. If K admits pullbacks along fibrations, the 2-functor cod : Fib(K) — K, which takes a fibration to its base, is a 2-fibration just like our original example over Cat.
where the the 2-cell \ corresponds to a comma-object, the morphism q is a fibration iff yy: q— q' has a right-adjoint in K/C. If K admits pullbacks along fibrations, the 2-functor cod : Fib(K) — K, which takes a fibration to its base, is a 2-fibration just like our original example over Cat.
where both squares on the left are pullbacks and the right and front are comma-objects. Since c is a cofibration, there is a left adjoint kK. 4 ne with (al ~\* Proof. Consider the diagram where both squares on the left are pullbacks and the right and front are comma-objects. Since c is a cofibration, there is a left adjoint kK. 4 n. with unit p: id > neke satisfying c'p = id (therefore K*n2(c’)* ee (c’)*). This adjunction implies that K% : Ex > Eel J is left biadjoint to n*, so that Kz ~ un.- By the same argument, 7 has a left adjoint &. with the same properties (namely, the pullback d*k_ of Kk along d). We have the following chain of equivalences
where both squares on the left are pullbacks and the right and front are comma-objects. Since c is a cofibration, there is a left adjoint kK. 4 ne with (al ~\* Proof. Consider the diagram where both squares on the left are pullbacks and the right and front are comma-objects. Since c is a cofibration, there is a left adjoint kK. 4 n. with unit p: id > neke satisfying c'p = id (therefore K*n2(c’)* ee (c’)*). This adjunction implies that K% : Ex > Eel J is left biadjoint to n*, so that Kz ~ un.- By the same argument, 7 has a left adjoint &. with the same properties (namely, the pullback d*k_ of Kk along d). We have the following chain of equivalences
where we have subsumed (and left nameless) composites of coherent isomorphisms associated to various reindexings of the 2-simplex o to simplify the diagram. For instance, the leftmost = is the composite
where we have subsumed (and left nameless) composites of coherent isomorphisms associated to various reindexings of the 2-simplex o to simplify the diagram. For instance, the leftmost = is the composite
forming a lax cocone for the unit and multiplication of the monad M and universal among such. compatible with the monad structure on M,N, while 2-cells a : (f,0) > (f’, 6’) are 2-cells a: f = f’ compatible with 6 and 6’. Forgetting the bimodules we get a domain/codomain object functor U : BiModr — R which has a left section Hom: R — BiModg sending an object X to its Hom-bimodule Homx: X 4“ X (whose top object is the object-of-arrows X ~). The existence of representable Kleisli objects for monads in Bimod(R) amounts to the exis- tence of a left biadjoint Q 4 Hom, in which case we get a 1-cell g: X — O(M) and a 2-cell
forming a lax cocone for the unit and multiplication of the monad M and universal among such. compatible with the monad structure on M,N, while 2-cells a : (f,0) > (f’, 6’) are 2-cells a: f = f’ compatible with 6 and 6’. Forgetting the bimodules we get a domain/codomain object functor U : BiModr — R which has a left section Hom: R — BiModg sending an object X to its Hom-bimodule Homx: X 4“ X (whose top object is the object-of-arrows X ~). The existence of representable Kleisli objects for monads in Bimod(R) amounts to the exis- tence of a left biadjoint Q 4 Hom, in which case we get a 1-cell g: X — O(M) and a 2-cell
The resulting 2-cell rn : gf = qg is the coinserter of f, g. We repeat this process working with bimodules rather than relations: given f,g: X — Y, we consider the ‘image’ bimodule f* e gz, compute the free monad on it, f* e gz and consider its representable Kleisli object: For instance in Cat(€), for € a topos with a natural numbers object, we could use the second construction above to compute coinserters. The construc- tion becomes particularly simple if the bimodule f* eg» already bears a monad structure, as it is the case when the pair f,g forms the (domain,codomain) data of a truncated 3-simplicial object. Notice that (Hermida, 2001, Prop. 8.3) shows that the construction of representable Kleisli objects for Cat(E) is fairly easy, requiring only pullbacks in the ambient category €.
The resulting 2-cell rn : gf = qg is the coinserter of f, g. We repeat this process working with bimodules rather than relations: given f,g: X — Y, we consider the ‘image’ bimodule f* e gz, compute the free monad on it, f* e gz and consider its representable Kleisli object: For instance in Cat(€), for € a topos with a natural numbers object, we could use the second construction above to compute coinserters. The construc- tion becomes particularly simple if the bimodule f* eg» already bears a monad structure, as it is the case when the pair f,g forms the (domain,codomain) data of a truncated 3-simplicial object. Notice that (Hermida, 2001, Prop. 8.3) shows that the construction of representable Kleisli objects for Cat(E) is fairly easy, requiring only pullbacks in the ambient category €.
Consider the diagram of pullbacks Since nf is right-adjoint right-inverse to Tp so is HF with respect to m, by (Hermida, 1999, Lemma 4.1). Proof. The first point to make is that y(p) : E[(*e)~'] > C is indeed a fibration since the monad B|_= id} : K/B — K/B preserves bounded coin- verters (by Proposition 5.5.(3)) and therefore the algebra structures of 75 and mp induce one on Ye (p).
Consider the diagram of pullbacks Since nf is right-adjoint right-inverse to Tp so is HF with respect to m, by (Hermida, 1999, Lemma 4.1). Proof. The first point to make is that y(p) : E[(*e)~'] > C is indeed a fibration since the monad B|_= id} : K/B — K/B preserves bounded coin- verters (by Proposition 5.5.(3)) and therefore the algebra structures of 75 and mp induce one on Ye (p).
which we have rewritten in terms of a ‘free-fibration’ comma-object pasted with a pullback square. Once again, we can factor x = 7,7, where zi, is the free cofibration and 7, has a right adjoint 7,. Hence pulling back along x, x* = nx(ni,)*, preserves the coinverter on 7p defining the direct image (pulling back along a cofibration does, and pulling back along 7, has a right biadjoint 7* when restricted to the full sub-2-category of the slices spanned by the fibrations). We thus get an appropriate coinverter on «* (app) = f’o(z*B), which is the one defining Uy ((z’)*p) since f’ is a fibration. For stability, consider a comma-object
which we have rewritten in terms of a ‘free-fibration’ comma-object pasted with a pullback square. Once again, we can factor x = 7,7, where zi, is the free cofibration and 7, has a right adjoint 7,. Hence pulling back along x, x* = nx(ni,)*, preserves the coinverter on 7p defining the direct image (pulling back along a cofibration does, and pulling back along 7, has a right biadjoint 7* when restricted to the full sub-2-category of the slices spanned by the fibrations). We thus get an appropriate coinverter on «* (app) = f’o(z*B), which is the one defining Uy ((z’)*p) since f’ is a fibration. For stability, consider a comma-object
where q is eso and f is fully-faithful and the diagonal d is essentially unique up to a unique isomorphism (compatible with a and 3) 5.10. Proposition. Essentially-surjective-on-objects are essentially orthogo- nal to fully faithful morphisms: given a pseudo-commutative diagram as on the left, it can be factorised as on the right
where q is eso and f is fully-faithful and the diagonal d is essentially unique up to a unique isomorphism (compatible with a and 3) 5.10. Proposition. Essentially-surjective-on-objects are essentially orthogo- nal to fully faithful morphisms: given a pseudo-commutative diagram as on the left, it can be factorised as on the right
In the pullback diagram above, both gp and gp are eso’s since they are pullbacks of q along fibrations. Since q*e is an equivalence, it is an eso, and therefore so is e: eq, and gp, eso imply e eso by Proposition 5.11. It remains to show that e is fully faithful. Consider a pair of morphisms a,b: 2— p in Fib(K)/Q and a 2-cell o : ea > eb. We get a 2-cell g*a : q*(ea) > q*(eb) and since q*(e) is fully faithful, there is a unique o’: q*(a) > q*(b) such g*(e)o’ = o. We thus have a 2-cell QpO’ : Adz = bz, and by the 2-dimensional universal property of the eso dx We get a unique 2-cell 6 : a > b such Gq, = gpa’. Also by universal- ity eo’ = a, and we conclude that e is fully faithful, and therefore an equivalence.
In the pullback diagram above, both gp and gp are eso’s since they are pullbacks of q along fibrations. Since q*e is an equivalence, it is an eso, and therefore so is e: eq, and gp, eso imply e eso by Proposition 5.11. It remains to show that e is fully faithful. Consider a pair of morphisms a,b: 2— p in Fib(K)/Q and a 2-cell o : ea > eb. We get a 2-cell g*a : q*(ea) > q*(eb) and since q*(e) is fully faithful, there is a unique o’: q*(a) > q*(b) such g*(e)o’ = o. We thus have a 2-cell QpO’ : Adz = bz, and by the 2-dimensional universal property of the eso dx We get a unique 2-cell 6 : a > b such Gq, = gpa’. Also by universal- ity eo’ = a, and we conclude that e is fully faithful, and therefore an equivalence.
A pseudo-coequaliser for a 2-fork consists of a pair (gq, a), where g : T > Q and a: qd => qc is an invertible 2-cell, such that pasting the appropriate 3 copies on the above subdivided 2-simplex yields a commutative cube:
A pseudo-coequaliser for a 2-fork consists of a pair (gq, a), where g : T > Q and a: qd => qc is an invertible 2-cell, such that pasting the appropriate 3 copies on the above subdivided 2-simplex yields a commutative cube:
and universal (up to equivalence) among such data (bicolimit property). We may regard the cube as a pseudo-natural transformation from the (diagram of the) top-face to the bottom one. We now seek a pseudo-splitting of this cube qua pseudo-natural transformation: it consists (in addition to the given pair (q, a) satisfying the equations above) of morphisms u: S > R,t:T — S and s:Q-—T, invertible 2-cells 0 : ct > sq, y : Tigu > tc and 6: mo2u => td satis- fying the equations to yield a cube as above, and invertible 2-cells vp : id > 71 vg : id => dt and vr: id > qs satisfying the equations of a modification.
and universal (up to equivalence) among such data (bicolimit property). We may regard the cube as a pseudo-natural transformation from the (diagram of the) top-face to the bottom one. We now seek a pseudo-splitting of this cube qua pseudo-natural transformation: it consists (in addition to the given pair (q, a) satisfying the equations above) of morphisms u: S > R,t:T — S and s:Q-—T, invertible 2-cells 0 : ct > sq, y : Tigu > tc and 6: mo2u => td satis- fying the equations to yield a cube as above, and invertible 2-cells vp : id > 71 vg : id => dt and vr: id > qs satisfying the equations of a modification.

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References (31)

  1. Bénabou, J. and R. Roubaud: 1970, 'Monades et descente'. C.R.A.S. Paris, A pp. 96-98.
  2. Blackwell, R., G. Kelly, and A. Power: 1989, 'Two dimensional monad theory'. Journal of Pure and Applied Algebra 59(1), 1-41.
  3. Borceux, F.: 1994, Handbook of Categorical Algebra II: Categories and Struc- tures, Vol. 51 of Encyclopedia of Mathematics and its applications. Cambridge University Press.
  4. Carboni, A., S. Johnson, R. Street, and D. Verity: 1994, 'Modulated bicategories'. Journal of Pure and Applied Algebra 94(3), 229-282.
  5. Day, B. and R. Street: 1997, 'Monoidal bicategories and Hopf algebroids'. Advances in Mathematics 129(1), 99-157.
  6. Freyd, P. J. and A. Scedrov: 1990, Categories, allegories, Vol. 39 of North-Holland Mathematical Library. Amsterdam: North-Holland Publishing Co.
  7. Gabriel, P. and M. Zisman: 1967, Calculus of Fractions and Homotopy Theory. Springer Verlag.
  8. Giraud, J.: 1964, 'Méthode de la descente'. Bull. Soc. Math. Fr., Suppl., Mm. 2. Gordon, R., A. J. Power, and R. Street: 1995, 'Coherence for tricategories'. Memoirs of the AMS 117(558).
  9. Grothendieck, A.: 1971, 'Catégories fibrées et descente'. In: A. Grothendieck (ed.): Revêtements étales et groupe fondamental, (SGA 1), Exposé VI, Vol. 224 of Lecture Notes in Mathematics. Springer Verlag.
  10. Hardie, K. A., K. H. Kamps, and R. W. Kieboom: 2002, 'Fibrations of bigroupoids'. J. Pure Appl. Algebra 168(1), 35-43.
  11. Hermida, C.: 1999, 'Some properties of Fib as a fibred 2-category'. Journal of Pure and Applied Algebra 134(1), 83-109. Presented at ECCT'94, Tours, France.
  12. Hermida, C.: 2001, 'From coherent structures to universal properties'. Journal of Pure and Applied Algebra 165(1), 7-61. preprint available math.CT/0006161.
  13. Hermida, C.: 2002, 'Fibrations and Yoneda structure for multicategories'. presented at Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, Fields Institute, Toronto, September 23-28.
  14. Janelidze, G. and W. Tholen: 1994, 'Facets of Descent I'. Applied Categorical Structures 2, 1-37.
  15. Janelidze, G. and W. Tholen: 1997, 'Facets of Descent II'. Applied Categorical Structures 5(3), 229-248.
  16. Joyal, A. and R. Street: 1993, 'Pullbacks equivalent to pseudopullbacks'. Cahiers Topologie Géom. Differentielle Catégoriques XXXIV(2), 153-156.
  17. Kelly, G.: 1989, 'Elementary observations on 2-categorical limits'. Bulletin Aus- tralian Mathematical Society 39, 301-317.
  18. Kelly, G. M.: 1974, 'Coherence theorems for lax algebras and for distributive laws'. In: Category Seminar (Proc. Sem., Sydney, 1972/1973). Berlin: Springer, pp. 281-375. Lecture Notes in Math., Vol. 420.
  19. Lack, S.: 2000, 'A coherent approach to pseudomonads'. Advances in Mathematics 152, 179-202.
  20. Le Creurer, I. J., F. Marmolejo, and E. M. Vitale: 2002, 'Beck's theorem for pseudo- monads'. J. Pure Appl. Algebra 173(3), 293-313.
  21. MacLane, S.: 1998, Categories for the Working Mathematician. Springer Verlag. 2nd. edition.
  22. Makkai, M.: 1993, 'Definability and Duality in First-Order Logic'. Vol. 503 of Memoirs of the AMS.
  23. Marmolejo, F.: 1999, 'Distributive laws for pseudomonads'. Theory and Applications of Categories 5, 91-147. available at http://www.tac.mta.ca/tac/.
  24. Moerdijk, I.: 1990, 'Lectures on 2-dimensional groupoids'. Séminaire de
  25. Mathématique, Institut de mathématique pure et appliquée, Université catholique de Louvain (175), 301-353.
  26. Moerdijk, I. and J. J. C. Vermeulen: 2000, 'Proper maps of toposes'. Mem. Amer. Math. Soc. 148(705), x+108.
  27. Street, R.: 1973, 'Fibrations and Yoneda's Lemma in a 2-category'. In: Category Seminar, Vol. 420 of Lecture Notes in Mathematics. Springer Verlag.
  28. Street, R.: 1980, 'Fibrations in bicategories'. Cahiers Topologie Géom. Differentielle Catégoriques 21.
  29. Street, R.: 1982, 'Two Dimensional Sheaf Theory'. Journal of Pure and Applied Algebra 23, 251-270.
  30. Street, R.: 1995, 'Descent Theory'. notes of a talk delivered at Higher descent structures, Geometric and Logical Aspects of Descent Theory 17.09.-23.09.1995 (Tagungsbericht 38/1995, Mathematisches Forschungsinstitut Oberwolfach).
  31. Street, R. and R. Walters: 1978, 'Yoneda structures on a 2-categories'. Journal of Algebra 50, 350-379.

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