## STRICT ∞-CATEGORIES. CONCRETE DUALITY (2006)

### BibTeX

@MISC{Kondratiev06strict∞-categories.,

author = {G. V. Kondratiev},

title = {STRICT ∞-CATEGORIES. CONCRETE DUALITY},

year = {2006}

}

### OpenURL

### Abstract

Abstract. An elementary theory of strict ∞-categories with application to concrete duality is given. New examples of first and second order concrete duality are presented. 1. Categories, functors, natural transformations, modifications There are two kinds of weakness happenning to ∞-categories. One is changing all occurences of equality = with a weaker equivalence realtion ∼. The other one is a weak naturality condition. The first one is not proper and implies strict category theory. The second one is proper and gives a weak category theory. Below we use ∼ instead of =. It is not necessary but has an advantage to treat directly the classification problem (up to ∼). Definition 1.1. • ∞-precategory is a (big) set L endowed with (1) grading L = ∐ Ln n≥0 (2) unary operations d, c: ∐ (3) unary operation e: ∐ n≥1 L n → ∐ L n → ∐ L n≥0