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A stack is an object that mixes the notions of (algebraic) space and group. The key insight of stack theory is that most things you would want to do with spaces you can do with stacks: namely, you hav …

Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or …

Suppose that we have a "diagram of categories", i.e. a map from some small category $J$, viewed as a 2-category (with trivial two-morphisms) to the category of categories, $j\mapsto \mathcal{C}_j.$ Th …

In the "ordinary" operad category, it is known that there is a colored operad $Op$ with set of colors $\mathbb{N}$ corresponding to "degrees" of vertices and with operations indexed by trees, such tha …

A wheeled properad is roughly, if I understand correctly, a properad (or PROP) with contraction maps $O_i^j\to O_{i-1}^{j-1}$ which contract an input with an output. There is a book, Infinity Properad …

I know of three homotopy theories of colored operads. The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with wea …

Say $\pi: C\to J$ is an inner fibration of $\infty$-categories. Then "morally", $\pi$ corresponds to a diagram indexed by $J$ in the "category of categories with correspondences", and if $\pi$ is coCa …

Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also …

It is my understanding that the $\infty$-category of non-unital connected topological monoids is equivalent to the $\infty$-category of connected topological groups. It follows that the functor from u …

Suppose $U$ is a (possibly singular) scheme and $X$ is a compactification (potentially unnecessary at least in characteristic $0$). Let $\pi:X\to *$ be the map to the point (though one can consider mo …

Let $k$ be a field of characteristic $p$ and $D^b(k)$ be the infinity (equivalently, DG) category of perfect complexes over $k$. Let $C_p(=\mathbb{Z}/p)$ be the cyclic group on $p$ elements. For a $C_ …

$\newcommand{\C}{\mathcal{C}}$ Suppose $M$ is a monad in an $\infty$-category $\C,$ and $A, B$ are two algebras over $M$. I'm willing to assume any reasonable "niceness" conditions on $\C$, $M$, etc: …

Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a functo …