TY - JOUR

T1 - Enriched ∞-categories via non-symmetric ∞-operads

AU - Gepner, David

AU - Haugseng, Rune

PY - 2015

Y1 - 2015

N2 - We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞-category. Our theory of enriched ∞-categories has many desirable properties; for instance, if the enriching ∞-category V is presentably symmetric monoidal then is as well. These features render the theory useful even when an ∞-category of enriched ∞-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and (∞,n)-categories). This is analogous to the advantages of ∞-categories over more rigid models such as simplicial categories — for example, the resulting ∞-categories of functors between enriched ∞-categories automatically have the correct homotopy type.We construct the homotopy theory of V-enriched ∞-categories as a certain full subcategory of the ∞-category of “many-object associative algebras” in V. The latter are defined using a non-symmetric version of Lurie's ∞-operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric ∞-operads. While we may regard these “many-object” algebras as enriched ∞-categories, we show that it is precisely the full subcategory of “complete” objects (in the sense of Rezk, i.e. those whose spaces of objects are equivalent to their spaces of equivalences) that are local with respect to the class of fully faithful and essentially surjective functors. We also consider an alternative model of enriched ∞-categories as certain presheaves of spaces satisfying analogues of the “Segal condition” for Rezk's Segal spaces. Lastly, we present some applications of our theory, most notably the identification of associative algebras in V as a coreflective subcategory of pointed V-enriched ∞-categories as well as a proof of a strong version of the Baez–Dolan stabilization hypothesis.

AB - We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞-category. Our theory of enriched ∞-categories has many desirable properties; for instance, if the enriching ∞-category V is presentably symmetric monoidal then is as well. These features render the theory useful even when an ∞-category of enriched ∞-categories comes from a model category (as is often the case in examples of interest, e.g. dg-categories, spectral categories, and (∞,n)-categories). This is analogous to the advantages of ∞-categories over more rigid models such as simplicial categories — for example, the resulting ∞-categories of functors between enriched ∞-categories automatically have the correct homotopy type.We construct the homotopy theory of V-enriched ∞-categories as a certain full subcategory of the ∞-category of “many-object associative algebras” in V. The latter are defined using a non-symmetric version of Lurie's ∞-operads, and we develop the basics of this theory, closely following Lurie's treatment of symmetric ∞-operads. While we may regard these “many-object” algebras as enriched ∞-categories, we show that it is precisely the full subcategory of “complete” objects (in the sense of Rezk, i.e. those whose spaces of objects are equivalent to their spaces of equivalences) that are local with respect to the class of fully faithful and essentially surjective functors. We also consider an alternative model of enriched ∞-categories as certain presheaves of spaces satisfying analogues of the “Segal condition” for Rezk's Segal spaces. Lastly, we present some applications of our theory, most notably the identification of associative algebras in V as a coreflective subcategory of pointed V-enriched ∞-categories as well as a proof of a strong version of the Baez–Dolan stabilization hypothesis.

KW - math.AT

KW - math.CT

KW - 18D20, 18D50, 55P48, 55U35

U2 - 10.1016/j.aim.2015.02.007

DO - 10.1016/j.aim.2015.02.007

M3 - Journal article

VL - 279

SP - 575

EP - 716

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -