Enriched mapping spaces via necklaces

Necklaces are combinatorial gadgets originally introduced by Baues (who called them ‘cellular strings’) and popularised by Dugger-Spivak in their study of the mapping spaces of quasi-categories [2]. A necklace is a simplicial set built from a sequence of standard sequences that are glued at their endpoints, e.g.

$$T={\mathrm{\Delta }}^3\vee {\mathrm{\Delta }}^2\vee {\mathrm{\Delta }}^1\vee {\mathrm{\Delta }}^3$$

The category $𝒩ec$ of necklaces has many desirable properties. It is a Reedy category with a compatible monoidal structure [1]. Moreover it is a test category, hence the category $NSet$ of necklicial sets $𝒩e{c}^{op}\to Set$ models homotopy types. Consider the category $NCat$ of categories enriched in $NSet$. Then there exists a fully faithful left-adjoint $SSet\hookrightarrow NCat$ [3] from simplicial sets into necklicial categories. This fact was (implicitly) used in [2] to construct alternative models for the left-adjoint of the homotopy coherent nerve. In this talk I will explain how this result may be generalised to left-adjoints of other nerves.

Let $(𝒲,\otimes ,I)$ be a monoidal simplicial model category. In [4], I provide a general procedure for constructing nerve functors $N^D:𝒲Cat\to SSet$ induced by a strong monoidal diagram $D:𝒩ec\to 𝒲$, and give conditions for when its left-adjoint $L^D$ may be described explicitly. In work in preparation I show, under some compatibility conditions between $𝒲$ and $D$:

This includes the left-adjoints of the homotopy-coherent, cubical and differential graded nerves. I will give a sketch of the proof, which is based on [2]. The crucial step is a surprisingly elegent combinatorial argument for necklaces.

This result fits into a broader project joint with Wendy Lowen [3], where we put forth the category of quasi-categories in a monoidal category $𝒱$ as a model for $S𝒱$-enriched categories with a homotopically well-behaved monoidal structure. When $𝒱=Set$, this recovers the classical comparison of quasi-categories with simplicial categories. Time permitting, I will expand on this and explain work in preparation on the case where $𝒱=Mod(k)$, the category of modules over a commutative ring $k$. I construct a monoidal cofibration category $S_{\otimes }Mod{(k)}_{cof}$, the fibrant objects of which are in particular quasi-categories in $Mod(k)$. Moreover, the above theorem allows to show an equivalence of homotopy categories

$$Ho(dgCa{t}_{\ge 0})\cong Ho(S_{\otimes }Mod{(k)}_{cof})$$

with connective dg-categories (equivalently $SMod(k)$-enriched categories by the Dold-Kan correspondence).

Finally, in joint work with Giuseppe Leoncini, we are developing the case where $𝒱$ is cartesian monoidal and locally connected. Time permitting, I will explain this work in further detail as well.