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Yes. First, set $Y^* = X^*$. Then, compose the evaluation and coevaluation maps of $X$ with maps like $\operatorname{id_{X^*}} \otimes \sigma$.

This is false for $D(G)$, when $G$ is sufficiently complicated. For a finite group $G$, the representation category of $D(G)$ has irreducible objects parametrized by pairs $(g, V)$ where $g$ is a con …

An object over a scheme $T$ on the left is given by a decomposition of $T$ into a parametrized disjoint union $T_i$ of schemes, and a parametrized family of triples $(x_i, y_i, \phi_i)$, where $x_i$ i …

The universal map is not a map of formal groups without some extra condition. An easy class of counterexamples comes from completions of an affine group along a closed subscheme that does not contain …

The answers to your questions are essentially in the first paper you cite. The second paper has more information on finer structure, like torsion, but the basic properties are all we need. In the fi …

In general, we expect field theories to be described by some higher categorical structures, where bulk models are assigned objects (also called 0-morphisms), domain walls are assigned morphisms (also …

For the case of vector spaces graded by an abelian group (with braiding determined by an abelian 3-cocycle following Joyal-Street), this was done by Dong and Lepowsky in their 1993 book "Generalized V …

I can't precisely isolate the wrong step, but it might help to consider a concrete motivational example, where $X$ is the $xy$-plane, $N$ is a very small interval in the $x$-axis, and $P$ is the origi …

As a first step, it may help to choose some relatively simple examples of $X$ and $Y$. If you can find a characterization of objects in your category as representable presheaves among all presheaves, …

If you take the Pontryagin dual of the inverse system of cyclic groups $\mathbb{Z}/N\mathbb{Z}$ and projections, you get a system of cyclic groups $\frac{1}{N}\mathbb{Z}/\mathbb{Z}$ with inclusions in …

You can define a unit interval $I$ as a co-presheaf: the set of maps from $I$ to a connected scheme $X$ is the set of triples $(x,y,\phi)$, where $x$ and $y$ are geometric points in $X$, and $\phi$ is …

The answer to your question is "yes". We need the following properties of $X_1^{iso}$: The map $X_1^{iso} \to X_1$ given by $(f,g) \mapsto f$ on scheme-valued points is a monomorphism. This is str …

The group ring of any group $G$ yields a special case of Noah's answer, where $C$ is the monoidal category of $G$-graded vector spaces. I wrote this up in a blog post a few years ago.

Given a functor from a small category to $Set$, we can describe the colimit set as a quotient of the disjoint union of image sets by an equivalence relation arising from morphisms in the source catego …

The basic answer is essentially as Emerton described in the comment. The most commonly used topologies on schemes are Zariski, Nisnevich, étale, smooth, syntomic, fppf, and fpqc, and this list is tot …