nLab change of enriching category

Definition

Given two monoidal categories $(V_1,\otimes_1, I_1)$ and $(V_2, \otimes_2, I_2)$ used for enrichment in enriched category theory (for instance two Bénabou cosmoi), a lax monoidal functor between them

$$F \;\colon\; (V_1, \otimes_1, I_1) \longrightarrow (V_2, \otimes_2, I_2)$$

canonically induces a 2-functor

$$F_\ast \;\colon\; V_1 Cat \longrightarrow V_2 Cat$$

between categories of V-enriched categories sending a $V_1$-enriched category $\mathcal{C}$ to the $V_2$-enriched category $F_\ast(\mathcal{C})$ such that

1. $F_\ast(\mathcal{C})$ has the same objects as $\mathcal{C}$;

2. the hom-objects of $F_\ast(\mathcal{C})$ are the images of the hom-objects of $\mathcal{C}$ under $F$:

   $$F_\ast(\mathcal{C})(x,y) \;\coloneqq\; F\big(\mathcal{C}(x,y)\big)$$

3. the composition, unit, associator and unitor morphisms in $F_\ast(\mathcal{C})$ are the images of those of $\mathcal{C}$ composed with the structure morphisms of the lax monoidal-structure on $F$.

(Eilenberg-Kelly 65)

This operation $F_\ast$ is sometimes called change of enriching category or change of enriching base or just change of base (e.g. Crutwell 14, chapter 4, Riehl 14, lemma 3.4.3). But notice that, despite some vague similarity, this is different from base change of slice categories.

In the special case that $\mathcal{C}$ has a single object and is hence (if thought of as pointed by that object) equivalently a monoid object in $V_1$, this statement reduces to the statement that lax monoidal functors preserve monoids (this Prop.)

Generalizations and enhancements

• The operation of change of enriching category is functorial from MonCat to 2Cat. In particular, any monoidal adjunction $V_1\rightleftarrows V_2$ gives rise to a 2-adjunction $V_1 Cat\rightleftarrows V_2 Cat$ (and also for profunctors).

• $V$-enriched categories can be defined more generally when $V$ is a multicategory, and any functor $F:V_1\to V_2$ between multicategories induces a change of enrichment 2-functor. Note that a functor of multicategorise between the underlying multicategories of two monoidal categories corresponds to a lax monoidal functor, as in the original version above. The multicategorical version also includes change of enrichment between closed categories.

• A different generalization is when $V$ is a bicategory, yielding bicategory-enriched categories. Any lax functor of bicategories induces a similar functor between its 2-categories of enriched categories.

• When $V_1$ and $V_2$ are cocomplete monoidal categories (or locally cocomplete bicategories), so that the bicategories $V_i$-Prof of $V_i$-categories and $V_i$-profunctors exist, change of enrichment also induces a lax functor $V_1 Prof \to V_2 Prof$. To make this version functorial on $MonCat_{cocomplete}$, we need to consider $V_i Prof$ as double categories, yielding a functor from MonCat to DblCat?.

• Finally, a generalization subsuming all of these is that for any virtual double category $V$ we can construct another virtual double category $V Prof$, and this construction is functorial.

Examples

Example

For any a monoidal category $V$, the functor $V(I,-): V \to Set$ is lax monoidal, hence induces a 2-functor from $V Cat$ to $Cat$. This assigns to any $V$-enriched category, $\mathcal{C}$, its underlying ordinary category, usually denoted $\mathcal{C}_0$, defined by $\mathcal{C}_0(x,y) = V(I, hom(x,y))$.

Example

If $V$ is a cocomplete monoidal category, the functor $V(I,-)$ above has a left adjoint that takes a set $X$ to the copower of $X$ copies of $I$. The resulting 2-functor $Cat \to V Cat$ takes an ordinary category to the “free” $V$-category it generates. Such $V$-categories are used, for instance, in subsuming “conical” limits under enriched weighted limits.

By functoriality, the adjunction between these two functors gives rise to a 2-adjunction $Cat \rightleftarrows V Cat$.

Example

More generally, if an adjunction $F:V_1 \rightleftarrows V_2:U$ can be regarded as a free-forgetful adjunction, then the corresponding adjunction between enriched categories can also be so regarded. For instance, if $V_1 = Top$ and $V_2 = G Top$ for some topological group $G$, then the right adjoint $V_2 Cat \to V_1 Cat$ takes the “underlying topologically enriched category” of a category enriched in $G$-spaces (its morphisms are the fixed-point spaces of the original $G$-space-enriched category; we think of these as the “$G$-equivariant maps” while the original spaces consisted of not-necessarily-equivariant maps acted on by conjugation). Similarly, any category enriched in spectra has an underlying category enriched in spaces (whose hom-spaces are the 0-spaces of the original hom-spectra), any dg-category has an underlying Ab-category (whose morphisms are the degree-0 ones in the original category), and so on.

Example

The geometric realization and total singular complex? adjunction $Real:SSet \rightleftarrows Top : Sing$ between simplicial sets and topological spaces induces another adjunction $SSet Cat \rightleftarrows Top Cat$ between simplicially enriched categories and topologically enriched categories.

Example

If $V=$SSet or Top, then the set of connected components $\pi_0:V\to Set$ is lax monoidal. The resulting change of enrichment functor takes a $V$-category $C$ to its “naive homotopy category” $h C(x,y) = \pi_0(C(x,y))$ obtained by “identifying homotopic morphisms”.

Similarly, the fundamental groupoid functor $\Pi_1:V\to Gpd$ is lax monoidal, so any $V$-category has an underlying (2,1)-category.

Example

An enhancement of the last example is that if $V$ is any monoidal model category, then its homotopy category $Ho(V)$ comes with a lax monoidal functor $\gamma : V \to Ho(V)$. Thus any $V$-category $C$ has an underlying $Ho(V)$-enriched “homotopy category” $h C$. Usually the underlying ordinary category of $h C$ is the naive homotopy category from the previous example.

Example

The nerve functor $N:Cat \to SSet$ preserves products and has a left adjoint $\tau_1$ that also preserves products. Thus, we have a change of enrichment adjunction in which any 2-category can be regarded as a simplicially enriched category, and similarly any simplicially enriched category has a “homotopy 2-category”. The latter plays an important role in the theory of quasi-categories.

Example

(poly-morphisms)

For $F = P \;\colon\; Set \to Set$ the power set-functor, the change of base functor $P_\ast \;\colon\; Cat \to Cat$ sends plain categories to plain categories. For $C$ any category, the morphisms of $C^{poly} \coloneqq P\ast(C)$ are called the poly-morphisms of $C$ in Mochizuki 12, section 0.

References

• Samuel Eilenberg, Max Kelly, Closed categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer (1966) [doi:10.1007/978-3-642-99902-4_22]

• Geoff Cruttwell, chapter 4 of: Normed spaces and the Change of Base for Enriched Categories, 2014 (pdf)

• Emily Riehl, lemma 3.4.3 inL Categorical Homotopy Theory, Cambridge University Press (2014)