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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

4 votes

Monad induced by actegory

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 …
John Gowers's user avatar
4 votes

It looks so coKleisli, but it's not. What is it?

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 …
John Gowers's user avatar
18 votes
3 answers
1k views

A multicategory is a ... with one object?

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 …
John Gowers's user avatar
15 votes
2 answers
687 views

Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C...

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 …
John Gowers's user avatar
3 votes
0 answers
117 views

What is the structure required to construct this homotopy of maps between mapping cones?

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 …
John Gowers's user avatar
4 votes
1 answer
317 views

What is the name for a natural transformation that has both lax and oplax monoidal properties?

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 …
John Gowers's user avatar
2 votes
0 answers
145 views

What name can I use for a cocone over the mapping cone diagram?

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. …
John Gowers's user avatar
3 votes
0 answers
201 views

What is the name of this construction on monoidal categories?

$\newcommand{\C}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\F}{\mathcal F} \renewcommand{\H}{\mathcal H} \newcommand{\from}{\colon} \newcommand{\tensor}{\otimes} \require{AMScd}$ Given mono …
John Gowers's user avatar
7 votes
0 answers
306 views

Is there a more general obstruction to the existence of moduli spaces than the existence of ...

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 …
John Gowers's user avatar