Terminology for a notion of "categories parameterized by another (symmetric monoidal) category" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:57:13Z http://mathoverflow.net/feeds/question/107143 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107143/terminology-for-a-notion-of-categories-parameterized-by-another-symmetric-monoi Terminology for a notion of "categories parameterized by another (symmetric monoidal) category" Yuji Tachikawa 2012-09-14T03:29:11Z 2012-09-14T06:09:31Z <p>Consider the category $Space(G)$ of $G$-spaces. For $H\subset G$, there is a forgetful functor from $Space(G)$ to $Space(H)$. Also, for an object $X$ in $Space(G)$ and another object $X'$ in $Space(G')$, $X\times X'$ is in $Space(G\times G')$.</p> <p>Consider also the category of $Mod(G)$ of modules over $H_G(pt)$. For $H\subset G$, there is a forgetful functor from $Mod(G)$ to $Mod(H)$. Also for a module $M$ in $Mod(G)$ and another module $M'$ in $Mod(G')$, $M\otimes_{\mathbb{Z}} M'$ is in $Mod(G\times G')$.</p> <p>My question is whether there is an established terminology for the categories $C(G)$ parameterized by a group $G$ satisfying the axioms abstracting those listed above:</p> <ul> <li>For $H\subset G$ there is a forgetful functor from $C(G)$ to $C(H)$</li> <li>For $G$, $G'$ there is a "multiplication functor" from $C(G)\times C(G')$ to $C(G\times G')$</li> </ul> <p>satisfying various relations. The equivariant cohomology functor $H_G: Space(G) \to Mod(G)$ is compatible with these two operations. Is there a terminology for such a functor?</p> <hr> <p>Update: Pondering more about it, I think it is better to formulate the concept as "categories $\mathcal{C}$ contravariantly parameterized by another (symmetric) monoidal category $\mathcal{X}$ ", i.e. </p> <ul> <li>for an object $X$ in $\mathcal{X}$, there is a category $\mathcal{C}(X)$</li> <li>for a morphism $f:X\to Y$ between objects in $\mathcal{X}$, there is a functor $f:\mathcal{C}(Y)\to\mathcal{C}(X)$</li> <li>for an object $X$ and $Y$ with its product $X\times Y$ in $\mathcal{X}$, there is a multiplication functor which defines, by $o_1\in \mathcal{C}(X)$ and $o_2\in\mathcal{C}(Y)$, an object $o_1\times o_2 \in \mathcal{C}(X\times Y)$.</li> </ul> http://mathoverflow.net/questions/107143/terminology-for-a-notion-of-categories-parameterized-by-another-symmetric-monoi/107147#107147 Answer by Mike Shulman for Terminology for a notion of "categories parameterized by another (symmetric monoidal) category" Mike Shulman 2012-09-14T06:09:31Z 2012-09-14T06:09:31Z <p>Zhen is right that you can think of it as a lax monoidal pseudofunctor. In <a href="http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html" rel="nofollow">this paper</a>, I called an equivalent structure a "monoidal fibration".</p>