Let $\mathcal{V}$ be a closed monoidal category, and $\mathscr{C}$ be a category enriched over $\mathcal{V}$. One says that the power or cotensor of an objec $A \in \mathscr{C}$ by an object $U \in \mathcal{V}$ is an object $A^U$ of $\mathscr{C}$ with a natural isomorphism $$\mathscr{C}(B, A^U) \cong \mathcal{V}(U, \mathscr{C}(B, A)),$$ where $B \in \mathscr{C}$, and $\mathcal{V}(-, -)$ is the inner hom of $\mathcal{V}$.

Riehl and Verity, in their third of a series of papers towards an axiomatic theory of $(\infty, 1)$-categories, define an $\infty$-cosmos to be some sort of category, the details of which do not really matter to my question. My question does, however, regard one certain part of their definition. They require that the class of isofibrations be closed under forming the Leibniz cotensor of two maps, and I am having a bit of difficulty seeing how this map is defined. Specifically, the Leibniz cotensor of a map $p: E \to B$ in $\mathscr{C}$ by a map $i: U \to V$ in $\mathcal{V}$ is a map

$$ i\, \widehat{\pitchfork}\, p: E^V \to E^U \times_{B^U} B^V. $$

Certainly, this map would follow by the universal property of pullbacks given maps $E^V \to E^U$ and $E^V \to B^V$ that satisfy the requisite commutativity condition. My question is as follows:

How does one define these maps? Indeed, a good definition of these maps is necessary even to form the pullback.

Any insight would be greatly appreciated.


It's pretty simple, really: using the universal property of $A^U$, one may define maps $E^i: E^V \to E^U$ and $p^V: E^V \to B^V$, and similarly maps $B^i: B^V \to B^U$ and $p^U: E^U \to B^U$.

For example, if $I$ is the monoidal unit of $\mathcal{V}$, one has a canonical arrow $\theta_{E, V}: I \to \mathcal{V}(V, \mathcal{C}(E^V, E))$ mated to the name of the identity $I \to \mathcal{C}(E^V, E^V)$, and then one may form the composite

$$I \stackrel{\theta_{E, V}}{\to} \mathcal{V}(V, \mathcal{C}(E^V, E)) \stackrel{\mathcal{V}(i, 1)}{\to} \mathcal{V}(U, \mathcal{C}(E^V, E)) \cong \mathcal{C}(E^V, E^U)$$

which names the map $E^i: E^V \to E^U$. The other arrows mentioned above may be constructed by similar considerations.

It is not hard to see by the yoga of enriched universal mapping properties that the cotensor thus becomes a $\mathcal{V}$-functor $\mathcal{V}^{op} \otimes \mathcal{C} \to \mathcal{C}$ in two variables, meaning in particular that there is a commutative square

$$\begin{array}{ccc} E^V & \stackrel{p^V}{\to} & B^V \\ E^i \downarrow & & \downarrow B^i \\ E^U & \underset{p^U}{\to} & B^U \end{array}$$

and so there is an induced map $E^V \to E^U \times_{B^U} B^V$ which is this thing you are asking about.

  • $\begingroup$ Thank you for your response. Would you be able to point me toward a reference on enriched category theory? I apologize if this MO isn't the right place for this; I didn't think my question would get any attention on math.se. $\endgroup$ – Reuben Stern Feb 27 '17 at 16:19
  • $\begingroup$ The classic reference I know of is Kelly's book Basic Concepts of Enriched Category Theory, available as a TAC reprint: tac.mta.ca/tac/reprints/articles/10/tr10abs.html. This certainly covers the types of issues encountered here. (By the way, you're probably right that this question would sink into deep layers of the stack at Math.SE.) $\endgroup$ – Todd Trimble Feb 27 '17 at 18:14

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