# Why Faonte called "small" and "big" dg-nerves?

I read G.Faonte "Simplicial nerve of an $$A_\infty$$-category" (https://arxiv.org/abs/1312.2127). In his paper, he calls two dg-nerve construction "small" and "big".

"Small" one means an $$\infty$$-category via (ordinal) dg-nerve of a dg-category. [HA. section.1.3.1] "Big" one means an $$\infty$$-category via Dold-Kan corrrespondence for each map space. [HA. section.1.3.1] (In fact, these $$\infty$$-categories are equivlent. he use big one for later proof.)

I don't understand why he uses terminology small and big? I cannot find his comments for this.

Can the latter construction(=big) use for big dg-categories(=map space is not small)?

• What is [HA.]? And some more characters. Commented May 24 at 0:10
• [HA] presumably means: J. Lurie, 2017, Higher algebra, (prepublication book draft), URL http: //www.math.harvard.edu/~lurie/papers/higheralgebra.pdf. Commented May 24 at 6:01
• @TimPorter that link is broken, this one works: people.math.harvard.edu/~lurie/papers/HA.pdf, or use the IAS one (which I assume is the preferred version): math.ias.edu/~lurie/papers/HA.pdf Commented May 24 at 7:04
• Thanks. I was using a link from an old set of notes of mine and had forgotten that that address no longer worked. Commented May 24 at 7:48

To relate the two, think of a simplicial model of a square, $$\Delta[1]\times \Delta[1]$$. That contains the diagonal 1-simplex. If one generates the free simplicial abelian group on that it has 5 non-degenerate generators in dimension 1, and two in dimension 2, now apply Dold-Kan and write down the corresponding chain complex. If, on the other hand, you first go across to chain complexes and then use the tensor product there is no diagonal and only one top dimensional generator. The Alexander-Whitney and Eilenberg-Zilber maps link the two constructions. The tensor product is smaller than the one coming from the product structure. This extends to the relation between the two versions of nerve.
The question of 'big' and 'small' dg-categories is a separate issue more related to set theoretic problems. The formal constructions that Faonte uses could be thought of as giving a 'big' simplicial class if applied to a 'big' dg-category. This just means that at certain points in the use of these ideas one may have to take a bit more care when doing certain types of construction, that is all. Big $$\infty$$-categories are as easy to handle as big dg-categories, or for that matter big categories in general. You can use either form of the dg- or $$A_\infty$$-nerve on a big dg or or $$A_\infty$$-category. It just needs a bit more care from time to time.