nLab (infinity,1)-categorification

Idea

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

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

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 $\mathrm{H}\mathbb{Z}$-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 $T$, one can construct an (∞,1)-categorification by passing to simplicial objects valued in algebras over $T$, 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? $\mathrm{Ani}(C)$ of a cocomplete category $C$ that is generated under colimits by its subcategory $C^{\mathrm{sfp}}$ of compact projective objects, is the (∞,1)-category freely generated by $C^{sfp}$ 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 $R$ is the (∞,1)-category of connective module spectra over the Eilenberg-MacLane ring spectrum $\mathrm{H}R$.

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
set	simplicial set up to a simplicial weak equivalence
groups	simplicial groups up to simplicial weak equivalences
algebras over algebraic theory $T$	simplicial algebras over $T$ up to simplicial weak equivalences
algebras over algebraic theory $T$	weak simplicial algebras over $T$ up to simplicial weak equivalences
abelian group	nonnegatively graded chain complex up to a quasi-isomorphism
abelian group	connective spectrum up to a weak equivalence
category	(∞,1)-category
category	relative category up to a Barwick-Kan equivalence
category	simplicial category up to a Dwyer-Kan equivalence
operad	(∞,1)-operad
operad	simplicial operad up to a weak equivalence
algebras over an operad $O$	simplicial algebras over a cofibrant resolution of $O$ up to a weak equivalence
Lie groupoid	Kan simplicial manifold
presheaf	simplicial presheaf
sheaf	(∞,1)-sheaf
sheaf	simplicial presheaf that satisfies the homotopy descent property
topos	(∞,1)-topos
elementary topos	elementary (∞,1)-topos
2-category	(∞,2)-category
n-category	(∞,n)-category