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Homotopic pullbacks, lax pullbacks, and exponentiability. (English) Zbl 1095.18001

A morphism \(q:Y\to B\) in a category \(\mathcal C\) is called exponentiable if pullbacks along \(q\) yield a left adjoint functor \(\mathcal C/B\to\mathcal C/B\). The paper analyzes what happens when pullbacks are replaced by homotopy pullbacks. The author mainly deals with topological spaces but other categories are considered as well.

MSC:

18A99 General theory of categories and functors
18B25 Topoi
18B30 Categories of topological spaces and continuous mappings (MSC2010)
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References:

[1] P.S. Alexandrov , Über die Metrisation der im kleinen kompakten topologische Räume , Math. Ann. 92 ( 1924 ), 294 - 301 . MR 1512216 | JFM 50.0128.04 · JFM 50.0128.04
[2] H.J. Baues , Algebraic Homotopy , Cambridge University Press , 1989 . MR 985099 | Zbl 0688.55001 · Zbl 0688.55001
[3] L. Bombelli , J. Lee , D. Meyer , and R. Sorkin , Space-time as a causal set . Phys. Rev. Lett. 59 ( 1987 ), 521 - 524 . MR 899046
[4] F. Conduché , Au sujet de l’existence d’adjoints à droite aux foncteurs ”image réciproque” dans la catégorie des catégories , C. R. Acad. Sci. Paris 275 ( 1972 ), A891 - 894 . MR 310033 | Zbl 0242.18012 · Zbl 0242.18012
[5] J. Giraud , Méthode de la descente , Bull. Math. Soc. France, Memoire 2 ( 1964 ). Numdam | MR 190142 | Zbl 0211.32902 · Zbl 0211.32902
[6] A. Grothendieck and J.L. Verdier , Theorie des Topos (SGA 4) , Springer Lecture Notes in Math. 269 - 270 ( 1972 ), 1 - 340 . MR 354653 | Zbl 0256.18008 · Zbl 0256.18008
[7] A. Grothendieck , Esquisse d’un programme , London Math. Soc. Lecture Note Ser. 242 ( 1997 ), 5 - 48 . MR 1483107 | Zbl 0901.14001 · Zbl 0901.14001
[8] K. A Hardie , K. H. Kamps , and T. Porter , The coherent homotopy category over a fixed space is a category of fractions , Topology Appl. 40 ( 1991 ), 265 - 274 . MR 1124841 | Zbl 0751.55010 · Zbl 0751.55010 · doi:10.1016/0166-8641(91)90109-Y
[9] J.M.E. Hyland , Function spaces in the category of locales , Springer Lecture Notes in Math. 871 ( 1981 ), 264 - 281 . Zbl 0483.54005 · Zbl 0483.54005
[10] J.R. Isbell , Atomless parts of spaces , Math. Scand. 31 ( 1972 ), 5 - 32 . MR 358725 | Zbl 0246.54028 · Zbl 0246.54028
[11] P.T. Johnstone , Topos Theory , Academic Press , 1977 . MR 470019 | Zbl 0368.18001 · Zbl 0368.18001
[12] P.T. Johnstone , Stone Spaces , Cambridge University Press , 1982 . MR 698074 | Zbl 0499.54001 · Zbl 0499.54001
[13] P.T. Johnstone and A. Joyal , Continuous categories and exponentiable toposes , J. Pure Appl. Algebra 25 ( 1982 ), 255 - 296 . MR 666021 | Zbl 0487.18003 · Zbl 0487.18003 · doi:10.1016/0022-4049(82)90083-4
[14] K.H. Kamps and T. Porter , Abstract homotopy and simple homotopy theory , World Scientific Publishing Co., Inc. , River Edge, NJ , 1997 . MR 1464944 | Zbl 0890.55014 · Zbl 0890.55014
[15] R.W. Kieboom , Notes on homotopy pull-backs , Quaest. Math. 14 ( 1991 ), 445 - 452 . MR 1143048 | Zbl 0747.55009 · Zbl 0747.55009 · doi:10.1080/16073606.1991.9631662
[16] M. Mather , Pull-backs in homotopy , Can. J. Math. 28 ( 1976 ), 225 - 263 . MR 402694 | Zbl 0351.55005 · Zbl 0351.55005 · doi:10.4153/CJM-1976-029-0
[17] I. Moerdijk and J.J.C. Vermeulen , Proper maps of toposes , Amer. Math. Soc. Memoirs 705 ( 2000 ). MR 1787303 | Zbl 0961.18003 · Zbl 0961.18003
[18] S.B. Niefield , Cartesianness , Ph.D. Thesis, Rutgers University , 1978 .
[19] S.B. Niefield , Cartesianness: topological spaces, uniform spaces, and affine schemes , J. Pure Appl. Algebra 23 ( 1982 ), 147 - 167 . MR 639571 | Zbl 0475.18011 · Zbl 0475.18011 · doi:10.1016/0022-4049(82)90004-4
[20] S.B. Niefield , Cartesian inclusions: locales and toposes , Comm. in Alg. 9 ( 16 ) ( 1981 ), 1639 - 1671 . MR 630579 | Zbl 0497.18009 · Zbl 0497.18009 · doi:10.1080/00927878108822672
[21] S.B. Niefield , Cartesian spaces over T and locales over \Omega (T) , Cah. Topol. Géom. Différ. Catég. 23 - 3 ( 1982 ), 257 - 267 . Numdam | Zbl 0492.18005 · Zbl 0492.18005
[22] S.B. Niefield , Exponentiable morphisms: posets, spaces, locales, and Grothendieck toposes , Theory Appl. Categ. 8 ( 2001 ), 16 - 32 . MR 1815044 | Zbl 1010.06010 · Zbl 1010.06010
[23] S.B. Niefield , Locally compact path spaces , Appl. Categ. Structures 13 ( 2005 ), 65 - 69 . MR 2132744 | Zbl 1076.54014 · Zbl 1076.54014 · doi:10.1007/s10485-004-5012-0
[24] D.P. Rideout and R.D. Sorkin , Evidence for a continuum limit in causal set dynamics , Phys. Rev. D 63 ( 2001 ), 104011 . MR 1840683
[25] D.S. Scott , Continuous lattices, Springer Lecture Notes in Math . 274 ( 1972 ), 97 - 137 . MR 404073 | Zbl 0239.54006 · Zbl 0239.54006
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