equivalences in/of $(\infty,1)$-categories
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.
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.
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 $\mathrm{H}R$.
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 |
Last revised on February 2, 2021 at 05:34:06. See the history of this page for a list of all contributions to it.