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Edit: Modified in accordance with Tom Leinster's entirely reasonable objections. Sorry to exhume this question from 5+ years ago. In case someone is still looking for an answer, note that a very spec …

Such questions are typically framed in terms of Classifying Spaces, but the answer is yes. It follows from, for instance, from Proposition 2.1 in Graeme Segal's article Classifying spaces and spectra …

I think what you have defined is just called the category of endomorphisms in $\mathcal{C}$. See for instance Marian Mrozek's 1992 paper Normal functors and retractors in categories of endomorphisms f …

Background: In homological algebra, a quasi-isomorphism of chain complexes is a chain map$\phi:(C,d) \to (C',d')$ so that the induced map on homology $\phi_\ast:H_\ast(C,d) \to H_\ast(C',d')$ is an is …

The following object has turned up in my research recently, and it would be surprising (to put it mildly) if no one else has seen, used or studied it before. I am hoping for a name and some references …

A recent project has forced me to consider a rather special object in a rather nasty category. Consider any category $\mathcal{C}$ which has image objects, meaning for each morphism $f: x \to y$ the …

A simplicial set $X$ is $k$-coskeletal iff the following condition holds: a simplex of dimension $\geq k$ is present iff all of its $(k-1)$-dimensional faces are present in $X$. A standard exa …

Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another …

Let $\mathcal{C}$ be a small category and $\Sigma$ a collection of morphisms in $\mathcal{C}$. Denote by $F_\Sigma:\mathcal{C} \to \mathcal{C}[\Sigma^{-1}]$ the usual quotient functor from $\mathcal{C …

A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context. Let $\mathcal{C}$ be a category. …

I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the pre …

I'm fairly confident that the following assertion is true (but I will confess that I did not verify the octahedral axiom yet): Let $T$ be a triangulated category and $C$ any category (let's say small …

The name of that article changed (a lot, it seems): the information you seek is in the paper Doctrinal Adjunction by Kelly. It lies on page 257 of the collection Category Seminar, Number 420 of Le …

On a more general note, you're (gradually) building a subcategory of the 2-fiber $\textbf{Cat}/C$. An excellent reference for those is the following paper, which helped me a lot when I was looking int …

Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a category of $T$-dynamical systems on $X$ which yields useful information about $X$ and $T$? More precisely, is there a category whose obj …