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Pairs of paths with the same source and target but with no other nodes in common are called parallel paths, at least on the computer science side of things in graph theory -- you can google the term t …

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 …

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 …

Quillen's Theorem A says that a functor $F:C \to D$ (between 1-categories) induces a homotopy equivalence of classifying spaces $BC \simeq BD$ if for every object $d$ in $D$ the fiber category $F/d$ h …

I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia: A module is a ring action on an abelian group. …

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 …

I'd been hoping for months that someone would come along and answer this question: every time I encounter the definition of microsupport, my brain responds with a flash of anger followed by a protract …

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 …

As mentioned in Carlsson and Zomorodian's paper (to which you have linked), the problem of computing persistence barcodes with coefficients in a ring $R$ relies essentially on classifying graded modul …

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 …

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 …

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 …

Here is a cool new (and very readable) preprint which uses the second barycentric subdivision (as discussed in Zhen Lin, Fernando Muro and Peter May's comments) to construct a cofibrantly generated mo …

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 $\mathbf{C}$ be a small category and $\mathbf{C}'$ its hammock localization in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion: If there is …