isomorphism isomorphism , weak equivalence , homotopy equivalence , weak homotopy equivalence , equivalence in an (∞,1)-category natural equivalence , natural isomorphism gauge

isomorphism isomorphism , weak equivalence , homotopy equivalence , weak homotopy equivalence , equivalence in an (∞,1)-category natural equivalence , natural isomorphism gauge

1)-category symmetric monoidal (∞,1)-category compact double category Category theory category theory Concepts category functor natural transformation Cat Universal constructions

category �� \mathcal{C} equipped with a braiding β \beta , which is a natural isomorphism β X , Y

1)-category dg-category A-∞ category triangulated category Morphisms k-morphism 2-morphism transfor natural transformation modification Functors functor

coefficients in A A is the set of connected components of the

1)-category dg-category A-∞ category triangulated category Morphisms k-morphism 2-morphism transfor natural transformation modification Functors functor

homotopy pushout homotopy realization homotopy coend Edit this sidebar Category theory category theory Concepts category functor natural transformation Cat Universal constructions

isomorphism isomorphism , weak equivalence , homotopy equivalence , weak homotopy equivalence , equivalence in an (∞,1)-category natural equivalence , natural isomorphism gauge

classification of topological phases Idea The notion of cohomology finds its natural general formulation in

inner product dual vector bundle stable vector bundle virtual vector bundle bundle of spectra natural bundle equivariant bundle Universal

theory of a manifold X X , there must also be a natural action of PU

topology sheaf sheafification quasitopos base topos , indexed topos Internal Logic categorical semantics internal logic subobject classifier natural numbers object Topos morphisms

degeneracy projections, def. satisfy the (dual) simplicial identities . Equivalently, they constitute the components of a

set Π 0 [ X , A ] \Pi_0[X,A] of connected path components of this

0 Pic ( R ) \pi_0 Pic(R) , but it is off only by H et 0 ( R , ℤ ) = ∏ components of R

isomorphism isomorphism , weak equivalence , homotopy equivalence , weak homotopy equivalence , equivalence in an (∞,1)-category natural equivalence , natural isomorphism gauge

category with weak equivalences by taking the weak equivalences to be those natural transformation s which are

inner product dual vector bundle stable vector bundle virtual vector bundle bundle of spectra natural bundle equivariant bundle Universal

cohomology (e.g. singular cohomology ) may be axiomatized by a small set of natural conditions, called the

1)-category symmetric monoidal (∞,1)-category compact double category Category theory category theory Concepts category functor natural transformation Cat Universal constructions

Transfors between 2-categories 2-functor pseudofunctor lax functor equivalence of 2-categories 2-natural transformation lax natural transformation

spaces ( CW-complexes ) to ℤ \mathbb{Z} - graded abelian groups (“ homology groups ”), in components E ˜ • : ( X ⟶ f

inner product dual vector bundle stable vector bundle virtual vector bundle bundle of spectra natural bundle equivariant bundle Universal

isomorphism isomorphism , weak equivalence , homotopy equivalence , weak homotopy equivalence , equivalence in an (∞,1)-category natural equivalence , natural isomorphism gauge

language relation between category theory and type theory Contents Idea Definition in components Symmetric operads Colored operads

topology sheaf sheafification quasitopos base topos , indexed topos Internal Logic categorical semantics internal logic subobject classifier natural numbers object Topos morphisms

single variational direction to many. Where Hamilton's equation appears as a natural condition in symplectic

the tangent bundle of configuration space.) This comes equipped with a natural 2 2 - form ω

1)-category symmetric monoidal (∞,1)-category compact double category Category theory category theory Concepts category functor natural transformation Cat Universal constructions

vector space V V over K K of dimension n n (a natural number ), an orientation of

principal bundles We give here a definition of covariant derivatives that is natural in the

R ) (the torsion equivalence classes of the Brauer stack ) It is therefore natural to regard

time translations U ( a , 1 ) U(a,1) . Axiom The components ϕ i \phi_{i

1)-category dg-category A-∞ category triangulated category Morphisms k-morphism 2-morphism transfor natural transformation modification Functors functor

that must be added to the system to break it into components that do not interact

but see there), is a monoidal category �� \mathcal{C} equipped with a natural isomorphism B x , y

cosmos , multicategory , bicategory , double category , virtual double category Basic concepts enriched category enriched functor , profunctor enriched natural transformation enriched adjoint functor

between (co) simplicial group s and (co) chain complex es. Both these categories carry natural monoidal category structures. It

inner product dual vector bundle stable vector bundle virtual vector bundle bundle of spectra natural bundle equivariant bundle Universal

A ) H(X,A) is the set of connected components in the

smooth ∞-groupoids ( ∞-stacks over the site of smooth manifolds ). It is therefore natural to define

s but fail to be (finite dimensional) Lie group s. They do however have natural realizations as smooth ∞-group

isomorphism isomorphism , weak equivalence , homotopy equivalence , weak homotopy equivalence , equivalence in an (∞,1)-category natural equivalence , natural isomorphism gauge

inner product dual vector bundle stable vector bundle virtual vector bundle bundle of spectra natural bundle equivariant bundle Universal

field K K . This definiting unwinds to the following structure in components Definition A ( n

for which K 0 ( C ) K_0(C) is the set of connected components C ↦ K ( C ) → π

dim = 2 dim = 2 : K3 surface generalized Calabi-Yau manifold Contents Idea Definition In components As functions on lattices

covariant phase space” is to indicate that it is obtained without any (necessarily un- natural , hence in physics

bi - graded objects from the homology/homotopy of the two graded components. Notably there is a

B G g : C(U) \to \mathbf{B}G is given in components precisely by a

been introduced and is often considered. But this formulation is more restrictive than is natural. Namely it is unnatural

to be invariant. With the above this now implies that the components of G

mechanism where the effective gauge fields in 4-dimensional spacetime arise as components of the

out that on cocycles of super Lie n-algebras there is a natural higher Lie theoretic operation