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This is close to the subject of my (so far unfinished) thesis, so I'll try to explain for the benefit of future readers. We define an oplax action of a monoidal category $\mathcal X$ upon a categor …

$\newcommand{\C}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\F}{\mathcal F} \renewcommand{\H}{\mathcal H} \newcommand{\from}{\colon} \newcommand{\tensor}{\otimes} \require{AMScd}$ Given mono …

In homotopy theory, the mapping cone of a continuous map $f\colon X \to Y$ is the homotopy pushout over the following span: $$ \require{AMScd} \begin{CD} X @>{f}>> Y\\ @VVV \\ \{*\} \end{CD} $$ I.e. …

Let $\mathcal C,\mathcal D,\mathcal E$ be monoidal categories, let $g$ be an oplax monoidal functor from $\mathcal C$ to $\mathcal D$ and let $G$ be a lax monoidal functor from $\mathcal D$ to $\mathc …

Let $X,Y$ be topological spaces, let $f$ be a continuous map from $X$ to $Y$ and let $g$ be a continuous map from $Y$ to $X$. Write $C_f$ for the mapping cone of $f$; i.e., $\{*\} + X\times I + Y$, w …

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a non-trivi …

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid in $[\mathcal C, \mat …

I'm going over what's already in the comments a little here, but: given an action of $C$ on $D$ as described, any monoid in $C$ gives rise to a monad on $D$. This is because a monoid in $C$ is just a …

We all know that A monoidal category is a bicategory with one object. How do we fill in the blank in the following sentence? A multicategory is a ... with one object. The answer is fairly …