On Comparing Definitions of "weak N–category"

1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand-and this is more relevant here-, I start "from scratch", and thus what I say can be understood with little technical knowledge. I only assume a modest amount of category theory as background. I will talk informally about technical matters that are written down formally elsewhere, where they can be studied further. [The text in square brackets [-] is either some technical explanation, or a digression.] 2. Terminology First, some terminological conventions. I will use the word "category" in its most general sense: weak ω-category. This is completely inclusive: all sorts of "categories" are categories ow. here are two extensions of the original meaning: "weak", and "omega-dimensional". Weak" signifies an indeterminate notion; there are several different specific versions of weak ategory. It can also be used as a vague notion, when one is merely looking at what one would ike to have. There are specific kinds of category, such as "Batanin category" [B1], "multitopic ategory" [HMP1,2,3,4,5], [M8]. When one wants to talk about the "ordinary, strict" version of he notion, one says "strict". Thus, "strict category" is my term for an ordinary, strict ω-category. he main reason for the terminology is the desire to banish the word "weak".-categories, for finite n , are "truncated" categories; they are particular kinds of, possibly eak, categories (namely, ω-categories) in fact.