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

Question

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)?

Answer 1

He uses big and small because the one he calls small is smaller!

Trying things out with a dg-category (rather than the more complicated A-infinity structure), the big one goes via the simplicially enriched category structure (which uses a product in the specification of the composition) so converts the dg-category structure to a S-categorical one. The small one uses more directly the tensor product of chain complexes in the construction (although this is partially hidden).

It may help to look at Lurie's Kerodon version of his construction and in particular https://kerodon.net/tag/00ND and https://kerodon.net/tag/00S0. The Alexander-Whitney map relates the free chains on a product simplicial set with the tensor product of the two parts. The first of these is 'big', and the second is 'small'.

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.

By the way, it is also worth glancing at the older version of the dg-nerve given in V.A.Hinich and V.V.Schechtman, 2006, On homotopy limit of homotopy algebras, in K-Theory, Arithmetic and Geometry: Seminar, Moscow University, 1984–1986, 240–264 (https://doi.org/10.1007/BFb0078370), to see another aspect of the constructions.