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I'll be rather general here, but I do not think the question is imprecise. It is known that given a cotriple $\bot:\mathcal{C}\to\mathcal{C}$ we can construct a canonical simplicial object of $\mathc …

This question begins with a sort of mysterious comment at the bottom of this Wikipedia page on injective cogenerators. There, it is said, without citation or proof, that as a result of the Tietze Exte …

Suppose given a well-generated triangulated category with a compatible symmetric monoidal structure, $\mathcal{T}$ (in the sense of Neeman). Is it clear that the image of a localization functor will …

Perhaps I should have been a little better at Googling before posting this question, but it seems to be answered, to a degree, in a paper from 1980 by Giuli, cited below. In particular, any epi-reflec …

In an answer to this question: Enriched slice categories, a description of the enrichment of the slice category in an enriched category is given. I'm interested in going a bit further. If we assume th …

We can think of the "tangent category" of a category $\mathcal{C}$ at an object $A$ as being the abelian group objects in the overcategory $\mathcal{C}/A$ (with whatever conditions I need on $\mathcal …

There is a forgetful functor from the category of (small) Cartesian monoidal categories (a symmetric monoidal category in which the tensor product is given by the categorical product) to the category …

In some approaches to operads and properads, categories of trees and graphs are used as "indexing categories" for this structure (see, for instance, https://arxiv.org/abs/0902.1954 or https://arxiv.or …

Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular …

Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a simp …

Answering this question took me some time. First of all, Liang Ze Wong and I had to write down a version of the enriched Grothendieck construction that worked for these purposes, as Tamaki's construct …

This question is answered in the affirmative in a recent paper of mine with Liang Ze Wong. In fact, we prove it more generally for a (strictly) monoidal simplicially enriched category. As any simplici …

There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated. Give some category $\mathcal{C}$ with the …

Here by stable homotopy category I mean the homotopy category of spectra, or maybe just some monogenic, Brown, algebraic, etc. stable homotopy category (in the language of Hovey, Palmieri and Strickla …

Is it known whether or not the endomorphism rings (or ring spectra) of the localized sphere spectra in $L_nSp$ and $L_{K(n)}Sp$ are Noetherian or not? Are they well understood? Perhaps, in the vein of …