Топологиялық кеңістіктер санаттары және категориялық топология

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N. B. Some authors use the name Top for the categories with topological manifolds, with
compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces.

The monomorphisms in Top are the injective continuous maps, the epimorphisms are the surjective continuous maps, and the isomorphisms are the homeomorphisms. The extremal monomorphisms are (up to isomorphism) the subspace embeddings. In fact, in Top all extremal monomorphisms happen to satisfy the stronger property of being regular. The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular. The split monomorphisms are (essentially) the inclusions of retracts into their ambient space. The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its retracts. There are no zero morphisms in Top, and in particular the category is not preadditive. Top is not cartesian closed (and therefore also not a topos) since it does not have exponential objects for all spaces. When this feature is desired, one often restricts to the full subcategory of compactly generated Hausdorff spaces CGHaus or the category of compactly generated weak Hausdorff spaces. However, Top is contained in the exponential category of pseudotopologies, which is itself a subcategory of the (also exponential) category of convergence spaces.

The category of pointed topological spaces Top• is a coslice category over Top. The homotopy category hTop has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category of Top. One can likewise form the pointed homotopy category hTop•. Top contains the important category Haus of Hausdorff spaces as a full subcategory. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with dense images in their codomains, so that epimorphisms need not be surjective. Top contains the full subcategory CGHaus of compactly generated Hausdorff spaces, which has the important property of being a Cartesian closed category while still containing all of the typical spaces of interest. This makes CGHaus a particularly convenient category of topological spaces that is often used in place of Top. The forgetful functor to Set has both a left and a right adjoint, as described above in the concrete category section. There is a functor to the category of locales Loc sending a topological space to its locale of open sets. This functor has a right adjoint that sends each locale to its topological space of points. This adjunction restricts to an equivalence between the category of sober spaces and spatial locales. The homotopy hypothesis relates Top with ∞Grpd, the category of ∞ groupoids. The conjecture states that ∞ groupoids are equivalent to topological spaces modulo weak homotopy equivalence.