algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle...

a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes...

the limit depends on the system of homomorphisms. Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual...

or inductive limit, and a limit becomes a colimit. We start with the definition of an inverse system (or projective system) of groups and homomorphisms...

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration...

order to more easily perform standard homotopy theoretic procedures such as homotopy colimits and homotopy limits, S h v N i s ( S m S ) {\displaystyle...

purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is...

all small limits and colimits exist in Top. In fact, the forgetful functor U : Top → Set uniquely lifts both limits and colimits and preserves them as well...

in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'...

finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales. If X {\displaystyle X} and Y {\displaystyle...