Compactly Generated Spaces and Quasi-spaces in Topology

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $$\textsf {Top}$$ Top of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class $$\mathcal {C}$$ C of objects we generalize the notion of $$\mathcal {C}$$ C -generated spaces , from which we derive, for instance, a general concept of Alexandroff spaces . Furthermore, as done for $$\textsf {Top}$$ Top , we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.

[1]  Dirk Hofmann,et al.  Probabilistic metric spaces as enriched categories , 2012, Fuzzy Sets Syst..

[2]  Martín Hötzel Escardó,et al.  A Constructive Model of Uniform Continuity , 2013, TLCA.

[3]  Dirk Hofmann,et al.  Lawvere Completion and Separation Via Closure , 2007, Appl. Categorical Struct..

[4]  Gavin J. Seal,et al.  Cartesian Closed Topological Categories and Tensor Products , 2005, Appl. Categorical Struct..

[5]  Reinhard Börger.  Coproducts and ultrafilters , 1987 .

[6]  Horst Herrlich,et al.  Abstract and concrete categories , 1990 .

[7]  Martín Hötzel Escardó,et al.  Topologies on spaces of continuous functions , 2001 .

[8]  Dirk Hofmann,et al.  Representable (T, V)-categories , 2014 .

[9]  Dirk Hofmann,et al.  Exponentiability in categories of lax algebras , 2003 .

[10]  Dirk Hofmann,et al.  Lawvere Completeness in Topology , 2007, Appl. Categorical Struct..

[11]  Eduardo J. Dubuc,et al.  Quasitopoi over a base category , 2006 .

[13]  Dirk Hofmann,et al.  Topological Features of Lax Algebras , 2003, Appl. Categorical Struct..

[14]  Dirk Hofmann,et al.  Monoidal topology : a categorical approach to order, metric, and topology , 2014 .

[15]  P. I. Booth.  A unified treatment of some basic problems in homotopy theory , 1973 .

[16]  R. Vogt.  Convenient categories of topological spaces for homotopy theory , 1971 .

[17]  Brian Day,et al.  A reflection theorem for closed categories , 1972 .

[18]  Nitakshi Goyal,et al.  General Topology-I , 2017 .

[20]  N. Steenrod,et al.  A convenient category of topological spaces. , 1967 .

[21]  D. Gale.  Compact sets of functions and function rings , 1950 .

[22]  E. Dubuc,et al.  Uniform spaces, spanier quasitopologies, and a duality for locally convex algebras , 1980, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[23]  Sarah L. Browne.  E-theory Spectra for graded C*-algebras , 2017, 1708.03258.

[24]  Dirk Hofmann.  The enriched Vietoris monad on representable spaces , 2012, 1212.5539.

[25]  E. G. Manes.  Compact Hausdorff objects , 1974 .

[26]  I. Petrakis.  Constructive topology of bishop spaces , 2015 .

[27]  J. Lawson,et al.  Comparing Cartesian closed categories of (core) compactly generated spaces , 2004 .

[28]  Dirk Hofmann,et al.  Cartesian closed exact completions in topology , 2018, Journal of Pure and Applied Algebra.

[29]  F. William Lawvere,et al.  Metric spaces, generalized logic, and closed categories , 1973 .

[30]  Dirk Hofmann,et al.  Injective Spaces via Adjunction , 2008, 0804.0326.

[31]  Dirk Hofmann,et al.  Exponentiation in V-categories , 2006 .

[32]  S. Lane.  Categories for the Working Mathematician , 1971 .

[33]  Dirk Hofmann,et al.  Topological theories and closed objects , 2007 .

[34]  L. D. Nel.  CARTESIAN CLOSED COREFLECTIVE HULLS , 1977 .

[35]  Dirk Hofmann,et al.  Effective Descent Morphisms in Categories of Lax Algebras , 2004, Appl. Categorical Struct..

[36]  Dirk Hofmann,et al.  On extensions of lax monads , 2004 .

[37]  TOPOLOGICAL PROPERTIES OF NON-ARCHIMEDEAN APPROACH SPACES , 2017 .

[38]  H. Herrlich,et al.  The quasicategory of quasispaces is illegitimate , 1983 .

[39]  B. Day.  Free finitary algebras on compactly generated spaces , 1981, Bulletin of the Australian Mathematical Society.

[40]  Walter Tholen,et al.  Metric, topology and multicategory—a common approach , 2003 .

[41]  Dirk Hofmann.  An algebraic description of regular epimorphisms in topology , 2005 .

[42]  Dirk Hofmann,et al.  The monads of classical algebra are seldom weakly cartesian , 2014 .

[43]  E-theory for C⁎-algebras over topological spaces , 2009, 0912.0283.

[44]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .