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Let $Idem = Idem^{(\infty)}$ be the walking idempotent [1], and let $Idem^{(n)}$ be its n-skeleton. Note that $Idem$ has one nondegenerate simplex in each dimension. Let $\iota_n^m: Idem^{(n)} \to Ide …

There are a number of formalisms available for presenting free strict $\omega$-categories -- Street's parity complexes, Steiner's directed complexes, computads, polygraphs,... Typically one has a cert …

Let $\mathcal C$ be an $\infty$-category. If $C$ is a quasicategory modeling $\mathcal C$, then we have a coend decomposition $$\mathcal C = \int^{[n] \in \Delta} \Delta[n] \times C_n.$$ This allows u …

Recall that a strict $n$-category $C$ is called gaunt if every $k$-morphism in $C$ with a weak inverse is an identity, for all $k$; let $Gaunt_n$ denote the strict 1-category of gaunt $n$-categories. …

Ross Street's 1991 paper Parity Complexes (apologies; I don't know how to find DOI links for Cahiers papers) develops some very useful tools for working with free strict $\omega$-categories. There is …

Let $\mathcal C$ be a sufficiently nice $\infty$-category, and let $f: U \to X$ be a morphism in $\mathcal C$. Recall that $f$ is said to be an effective epimorphism if the induced map $|U^{\times_X ( …

Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with t …

Question: What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D_i)_{i < ORD}$? In other words …

I'm interested in instances of the following data: $C$ is a (possibly higher) category; $(L,M)$ is a weak factorization system (wfs) on $C$; $(M,R)$ is a unique factorization system (fs) on $C$. …

Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim_{j \in J} I_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\var …

Let $\mathcal{M}$ be a model category presenting an $\infty$-category $\mathcal{C}$. I believe that every left Bousfield localization $\widetilde{\mathcal{M}}$ of $\mathcal{M}$ corresponds to a reflec …

Question 1: Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ als …

In the category of commutative rings, every domain embeds into a field. Is this true in the category of presentably symmetric monoidal stable $\infty$-categories? Here's what I mean by that. Let $Pr …

Question: What are some "surprising" examples of functors (resp. $\infty$-functors) $F$ which preserve cofiltered limits? I'm not quite sure what "surprising" means, but I think that "Surprising" sho …

If $\mathcal C$ is a $\kappa$-accessible 1-category, then the category of morphisms $Mor \mathcal C$ is a $\kappa$-accessible 1-category, with the $\kappa$-presentable objects being those morphisms wh …