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(infinity,1)-categorification

Contents

Contents

Idea

Categorification is a creative nonmechanical process in which categorical structures are promoted to nn-categorical structures, for some n2n\ge 2.

(∞,1)-categorification is a special instance of this idea, in which the newly created nn-morphisms are invertible for all n2n\ge 2.

The phrase higher structures also refers primarily to (∞,1)-categorification.

Sometimes more than one (∞,1)-categorification is possible, as is the case for abelian groups, which can be categorified to H \mathrm{H}\mathbb{Z} -module spectra (as represented by simplicial abelian groups) or connective spectra.

In addition to this creative choice of a Platonic form categorifying a given structure, another creative aspect is a choice of a specifc model for the resulting object, e.g., (∞,1)-categories can be modeled by relative categories, simplicial categories, quasicategories, etc.

Simplicial objects

For algebras over an algebraic theory TT, one can construct an (∞,1)-categorification by passing to simplicial objects valued in algebras over TT, and equipping them with weak equivalences induced by the forgetful functor to simplicial sets.

In some cases, the result can be different from the result of the animation procedure described below, e.g., for the algebraic theory that defines commutative monoids we get commutative simplicial monoids, equivalently, E-infinity algebras over the Eilenberg-MacLane spectrum of the integers, whereas animation produces connective E-infinity ring spectra.

Animation

In some cases, there is an automatic (∞,1)-categorification. For example, the animation? Ani(C)\mathrm{Ani}(C) of a cocomplete category CC that is generated under colimits by its subcategory C sfpC^{\mathrm{sfp}} of compact projective objects, is the (∞,1)-category freely generated by C sfpC^{sfp} under sifted colimits. (See Kęstutis Česnavičius and Peter Scholze, Sec. 5.1.4.)

For example, the animation of the 1-category of modules over an ordinary ring RR is the (∞,1)-category of connective module spectra over the Eilenberg-MacLane ring spectrum HR\mathrm{H}R.

Further Examples

In the table below, structures on the left are always understood up to an isomorphism, whereas on the right we explicitly indicate the notion of a weak equivalence used (except for Platonic forms such as (∞,1)-categories).

categorical structure(∞,1)-categorical structure
set∞-groupoid
setsimplicial set up to a simplicial weak equivalence
groupssimplicial groups up to simplicial weak equivalences
algebras over algebraic theory TTsimplicial algebras over TT up to simplicial weak equivalences
algebras over algebraic theory TTweak simplicial algebras over TT up to simplicial weak equivalences
abelian groupnonnegatively graded chain complex up to a quasi-isomorphism
abelian groupconnective spectrum up to a weak equivalence
category(∞,1)-category
categoryrelative category up to a Barwick-Kan equivalence
categorysimplicial category up to a Dwyer-Kan equivalence
operad(∞,1)-operad
operadsimplicial operad up to a weak equivalence
algebras over an operad OOsimplicial algebras over a cofibrant resolution of OO up to a weak equivalence
Lie groupoidKan simplicial manifold
presheafsimplicial presheaf
sheaf(∞,1)-sheaf
sheafsimplicial presheaf that satisfies the homotopy descent property

Last revised on February 2, 2021 at 05:34:06. See the history of this page for a list of all contributions to it.