11

For any $1$-category $C$ the localization $C[C^{-1}]$ at all arrows is an $\infty$-groupoid homotopy equivalent to the nerve of $C$, so it can be any $\infty$-groupoids.
For example take $C$ to be the poset with 6 elements ordered as a,b < c,d < e,f and when you localize at all arrows you get the $2$-sphere $\mathbb{S}^2$. So the simplicial ...

9

I'm going to assume that $\mathcal{V}$ is closed as well, or at least that its tensor product preserves colimits in each variable; I'm not sure that you get the left adjoint from $\mathsf{Cat}$ to $\mathcal{V}\text{-}\mathsf{Cat}$ otherwise. In this situation the underlying adjunction $\mathsf{Set} \rightleftarrows \mathcal{V}$ is monoidal, i.e. its left ...

8

The point is what Ivan hints at in his last paragraph, that additivity is a property rather than an extra structure.
In fact, we have:
Suppose $C$ is an additive category. Then the forgetful functor $Fun^\times(C,\mathbf{Ab})\to Fun^\times(C,\mathbf{Set})$ is an equivalence of categories.
(if you only assume $C$ to be preadditive, then the same holds with &...

rt.representation-theory ct.category-theory enriched-category-theory locally-presentable-categories accessible-categories

6

It is very easy to specify an answer if $P$ is a compact projective generator (you didn't write whether it is compact). Then, we have an equivalence of categories between $\mathcal{C}$ and the category of right modules over $\text{End} (P)$, given by sending $X$ to $\mathcal{C} (P , X)$. So, of course, if objects become isomorphic under an equivalence of ...

5

First of all, as David Roberts pointed out in a comment above, the composition map
c :: hom a b -> hom b c -> hom a c
c f g = g . f
and the evaluation map
ev :: (a -> b) -> a -> b
ev f x = f x
share a common property, that of forming the components of a cowedge. I will now switch to category-theory notation, I'm confident you'll be able to ...

4

The equivalence you mention holds more generally whenever your base of enrichment has a finitely presentable unit. This certainly includes $Ab$ but also many other examples: $Cat$, $sSet$, $GAb$, $DGAb$, etc.
Assume that $\mathcal V=(\mathcal V_0,\otimes,I)$ is a symmetric monoidal closed complete and cocomplete category. Then you can define the Ind-...

rt.representation-theory ct.category-theory enriched-category-theory locally-presentable-categories accessible-categories

4

Let $\mathcal{V}$ be a cocomplete monoidal category which can be presented by a limit theory, so that $\mathcal{V} = \mathsf{Lex}(\mathbb{T},\text{Set})$, of course this is the case of your question.
Now, let me give the answer under the assumption that $C$ has finite $\mathcal{V}$-enriched colimits. In this case the Ind-completion is described by the ...

rt.representation-theory ct.category-theory enriched-category-theory locally-presentable-categories accessible-categories

1

I think that the best answer so far comes from this paper: https://arxiv.org/abs/1811.07830 , where it is proved that homotopy categories of dg-categories and various flavours of A-infinity categories are equivalent.

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