On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$-algebra

In this paper, we develop the $$A_\infty $$ A ∞ -analog of the Maurer-Cartan simplicial set associated to an $$L_\infty $$ L ∞ -algebra and show how we can use this to study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of $$A_\infty $$ A ∞ -algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) $$A_\infty $$ A ∞ -algebras to simplicial sets, which sends a complete curved $$A_\infty $$ A ∞ -algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads.

W-types in homotopy-type theory – CORRIGENDUM

In the article below, Theorem 3.4 requires the additional assumption that A is Kan as well. Indeed, the inductive proof as given only shows that if W(f)<α is a Kan complex, then W(f)<α+1 → A is a Kan fibration.

The Ceramic Sequence at Nakbe, Guatemala

The site of Nakbe, located approximately 13 km southeast of El Mirador in the far northern part of the Peten, has been investigated by the RAINPEG Project, directed by Richard Hansen, for the last four field seasons. The ceramic sequence from Nakbe has provided us with a much broader view of cultural development in the north-central Peten. We have defined a series of preliminary ceramic complexes that span Middle Preclassic through Late Classic times.The earliest complex at Nakbe, called Ox, which belongs to the Mamom horizon, is one of the two best represented at the site, and definitely associated, late in the period, with large-scale architecture. The principal ceramic groups are the Juventud, Chunhinta, and Pital. The main forms are bowls or basins with flaring walls and direct or everted rims, short-necked jars, cuspidors or semicuspidors, and composite-silhouette bowls. Particularly notable, although rare, are the types Muxanal Red-on-Cream and Tierra Mojada Resist. The Ox Complex is characterized by a high frequency of decoration executed by penetration methods, especially incision and chamfering.The unslipped pottery pertains to the types Achiotes Unslipped and Palma Daub. The latter is marked by a red wash applied to the exterior neck of the jar, the major form in these types. Daub is a form of decoration limited to the Middle Preclassic in this area, as is the chamfering technique on the slipped pottery.The ceramic complexes most similar to Ox are located to the south of Nakbe at Uaxactun and Tikal. The high frequency of chamfering, daub, and other traits strongly link Nakbe to these southern sites during the Middle Preclassic, while sites to the north and southwest, such as Seibal, Altar de Sacrificios, and Becan exhibit more tenuous connections.The Kan Complex belongs to the Late Preclassic period, and is characterized by the Sierra, Polvero, Flor, and Sapote Ceramic Groups. Although associated with the large structures in the site center, Kan ceramics are less abundant than Ox ceramics in our samples. Kan pottery corresponds closely to that of El Mirador and is similar to other complexes of the Chicanel horizon. Particularly noteworthy are everted rims with circumferential grooves and lateral, labial, and medial flanges and ridges. The unslipped pottery of the Kan Complex consists almost exclusively of jars bearing exterior striation from the shoulder to the base.The closest ceramic ties to Kan pottery continues to be with the southern complexes, especially Uaxactun and Tikal. Moreover, the similarity to Seibal is greater during Chicanel times, while Belize appears to differentiate itself ceramically from the Peten during this time. Nevertheless, the Chicanel horizon is the period in which the maximum geographical extent of a ceramic sphere is reached.“Protoclassic” ceramics are rare, but the small amounts recovered at Nakbe are similar to those from El Mirador. The major type is Iberia Orange, found in small quantities in surface contexts. Characterized by hollow, mammiform supports, hooked rims, and orange slip, this Nakbe pottery seems to be more similar to pottery at Seibal than to other complexes with Protoclassic pottery.Early Classic pottery is virtually nonexistent in our excavations, suggesting an insignificant occupation during this period.Late Classic pottery (Uuc Complex) appears in significant quantities at Nakbe, mainly in the outskirts of the site. All of the types and modes defined at El Mirador are found at Nakbe, principally Tinaja Red, Chinja Impressed, Infierno Black, and Carmelita Incised, as well as the polychrome types. Moreover, Codex-style polychrome was also found at the site. As at El Mirador, the Uuc ceramics are not found in association with large-scale architecture, and it appears that Nakbe was not a major center at this time.The major occupations at Nakbe pertain to the Middle and Late Preclassic periods. The abundance of architecture, ceramics, and other classes of artifacts from the Ox Complex provides us with an opportunity to investigate a Middle Preclassic occupation in which there was a much more complex social organization, at least at Nakbe, than had previously been suspected.