MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question is motivated by my recent divertissements in the realm of nerves and realizations.

$\bf Set$-categories have the "classical nerve" sending $\bf C$ to the simplicial set $[n]\mapsto Fun([n],{\bf C})$; this could be the end of the story, but (un)fortunately categories can have much more structure: it would be stupid not to take into account the fact that $\bf C$ can be enriched over some other symmetric monoidal closed category $\cal W$, or can have additional structure we want to preserve (being a topos, being monoidal, being abelian, ...). Then we face various situations, letting $\bf A$ be from time to time a different kind of category which we want to send to a "coherently determined" simplicial set $N\bf A$:

  1. If ${\cal W}=\bf sSet$, we get Cordier's coherent nerve sending ${\bf A}\in {\cal W}\text{-}\bf Cat$ to $Fun({\frak C}[n],{\bf A})$, where ${\frak C}(-) \colon \Delta \to {\bf sSet}\text{-}\bf Cat$ sends [n] in a "standard" simplicial category; this is the "simplicial thickening".
  2. With suitable restrictions on $\bf Topoi$, we can get a "toposophic thickening" (click) of the nerve construction, which gives a simplicial set "naturally coherent" with the fact that $\bf A$ is not only a mere category, but a topos;
  3. If $\bf A$ is a dg-category as defined in Lurie's HA, ch I.1.3, then we can build th nerve $N_\text{dg}$, which albeit far from obviously arising from a cosimplicial object in the nerve-realization paradigm, seems to be "best-suited" to cope with the case $\bf A$ has an enrichment over chain complexes;
  4. The Duskin nerve provides a construction which is best-suited to cope with the case where $\bf A$ is monoidal: regard $\bf A$ as the one-object bicategory $B\bf A$, and then apply Duskin's machinery.

For the moment, forget about the "having more structure" case, which is unfortunately too difficult to axiomatize. (It's maybe possible to take a "doctrinal" point of view, but I don't want to get too far). My question is:

is there a "general theory" coping with the enriched case? Given a ${\cal W}$-category $\bf A$, what is the "canonical" way to define a nerve construction which "coherently takes into account" the $\cal W$-enrichment?

Partial answer: This seems to boil down to

  1. cocompleteness of ${\cal W}\text{-}\bf Cat$, to ensure that the equivalence $Fun(\Delta,{\cal W}\text{-}{\bf Cat})\cong {\rm Adj}(\widehat{\Delta},{\cal W}\text{-}{\bf Cat})$ holds true; is this cocompleteness automatic or it depends on some additional assumptions on $\cal W$?
  2. Presence of a "canonical" choice for a functor $\Delta\to {\cal W}\text{-}{\bf Cat}$, which generates by Kan extension a "realization", which has a right adjoint; again, I'm not sure I can find such a functor for any $\cal W$.

This right adjoint is precisely the "coherent nerve" I'm looking for.

share|cite|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.