In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit...

In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the...

(and pullback) sheaves in algebraic geometry, and pullback bundles in algebraic topology and differential geometry. See also: Pullback (category theory) Fibred...

functor Hom functor Product (category theory) Equaliser (mathematics) Kernel (category theory) Pullback (category theory)/fiber product Inverse limit...

the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as morphisms in a category of fractions. The notion...

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products...

morphism A ′ → A {\displaystyle A'\to A} , the pullback is an I {\displaystyle I} -indexed coproduct of the pullbacks: ( ∐ i ∈ I B i ) × A A ′ ≅ ∐ i ∈ I ( B i...

Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the...

In category theory, a branch of mathematics, a presheaf on a category C {\displaystyle C} is a functor F : C o p → S e t {\displaystyle F\colon C^{\mathrm...

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows...

a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.) Notes on foundations: In many expositions...

^{*}\left(\nabla _{d\phi (X)}s\right).} Pushforward (differential) Pullback bundle Pullback (category theory) Jost, Jürgen (2002). Riemannian Geometry and Geometric...

In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in...

limit – Construction in category theory Cartesian closed category – Type of category in category theory Categorical pullback – Most general completion...

terminology is contrary to the one used in category theory because it is the covectors that have pullbacks in general and are thus contravariant, whereas...

set theory but the general definition make for a richer range of logics. So consider an object Y {\displaystyle Y} in a category with pullbacks. Any...

object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. Category theory is a branch of mathematics that...

In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces...

In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified...

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures...

bundles. In the language of category theory, the pullback bundle construction is an example of the more general categorical pullback. As such it satisfies the...

In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such...

In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite...

projection p. The bundles on the Xij that we must control are Vi and Vj, the pullbacks to the fiber of V via the two different projection maps to X. Therefore...

In category theory, an end of a functor S : C o p × C → X {\displaystyle S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} } is a universal...

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general...

g) = Ker(f - g), where Ker denotes the category-theoretic kernel. Any category with fibre products (pullbacks) and products has equalisers. In Top where...

differential geometry Pullback (category theory), a term in category theory Pullback attractor, an aspect of a random dynamical system Pullback bundle, the fiber...

In category theory, a branch of mathematics, a pulation square (also called a Doolittle diagram) is a diagram that is simultaneously a pullback square...

scheme theory completely subsumes the theory of commutative rings. Since Z is an initial object in the category of commutative rings, the category of schemes...