# 2-completeness analog of completeness theorem

It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations".

I wonder if there is a 2-version of this. In particular,

Does a category have all finite 2-limits if it has all 2-equalizers and 2-products?

My instinct is no, and that we will need another(or several more) limits to build all 2-limits.

Of course the question can be generalized to n-limits, and I'm curious about that also.

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The nlab has an overview over some classes of 2-limits which do not arise for 1-categories, for example inserters. You cannot generate them from 2-equalizers or 2-products. – Martin Brandenburg Oct 12 '11 at 12:23

## 1 Answer

Your suspicion is correct: in general, a V-category has all weighted V-limits if it has all conical V-limits and is cotensored over V (see Kelly's Basic Concepts of Enriched Category Theory, section 3.10). For V = Cat (and this is true for bicategories too), cotensors can be constructed from conical limits and cotensors with the arrow category 2, although I don't know the original reference for that. 'Finite' limits are a bit more complicated in the enriched case, but see Street's 'Limits indexed by category-valued 2-functors', JPAA 1972, for the 2-case.

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I would add Kelly's "Structures defined by finite limits in the enrichment context" as a reference for finite enriched (co)limits. – Nacho Lopez Oct 12 '11 at 9:49
Yes, good point. That's here: numdam.org/item?id=CTGDC_1982__23_1_3_0 – Finn Lawler Oct 12 '11 at 16:20
A reference for the bicategorical case (meaning, for non-strict 2-limits -- I'm not sure whether the questioner meant to ask about strict ones or non-strict ones) is Street's correction to "Fibrations in bicategories": numdam.org/item?id=CTGDC_1987__28_1_53_0 – Mike Shulman Oct 19 '11 at 17:21