@SanathK.Devalapurkar just my ignorance here, but what does it mean for a map of stacks to be representable?
user105491
02:35
@JonBeardsley If $N\to M$ is a morphism of stacks then it's representable if for all affine maps $\mathrm{Spec}(R)\to M$, the fiber product $N\times_M\mathrm{Spec}(R)$ is equivalent to an affine scheme.
Hmmm... this seems weird... given an $E_n$-ring spectrum map $\phi:R\to S$ then is the induced functor $LMod_R\to LMod_S$, given by $S\otimes_R -$, $E_n$-monoidal, or only $E_{n-1}$-monoidal?
It's given by a relative tensor product construction, which, at least for quasicategories, I'm still a little shaky on the operadic properties of.
the reason I ask about directed HoTT is that i have been toying with the idea of thinking about a presentation of a monoid as a quotient of a free category, in particular we consider $F(X)$ as a directed homotopy type (i.e. just a category) where the points are strings with entries in $X$ (i.e. the elements of $F(X)$) and each object $t_i$ has paths $p_i: t_i \to x_i t_i$ corresponding to left multiplication by the generators
this realizes the factorizations of an element $t$ in the monoid, as directed paths from the base point (the identity element) to the point $t$
quite a different picture from the usual monoid-as-category-with-one-element
($X$ a set, $F: Set \to Mon$ the free monoid functor)
bad subscripting, should be each object $t_i$ has paths $p_{x_j}: t_i \to x_j t_i$ (for all $x_j \in X$)
the nlab doesn't suggest any survey papers, and an '09 MO answer didn't know of any either, but that might've changed by now. unfortunately it's kinda hard to force a google search to be relevant
random mostly-unrelated question: is there a name / universal characterization for the $\infty$-category obtained from all chain complexes (equipped with their canonical self-enrichment), without inverting the quasi-isomorphisms? this would of course be a stable + abelian analog of the "strong homotopy theory of simplicial sets"
@AaronMazel-Gee on the level of homotopy categories it's usually called the "homotopy category of chain complexes of R-modules" and called something like K(R). i don't know of many situations where "category X together with its simplicial enrichment" is called something different from "category X"
@TylerLawson sure, yeah people often use that notation K(R) for the "strict homotopy category". but i've only seen it defined as a homotopy category, and only on the way to defining the derived category. i'm more curious about uses / universal characterizations than names, i guess
oops haha sorry, somehow i misread and didn't notice that you already said you were referring to the homotopy category
