nLab
change of enriching category

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Definition

Given two monoidal categories (V 1, 1,I 1)(V_1,\otimes_1, I_1) and (V 2, 2,I 2)(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:(V 1, 1,I 1)(V 2, 2,I 2) F \;\colon\; (V_1, \otimes_1, I_1) \longrightarrow (V_2, \otimes_2, I_2)

canonically induces a 2-functor

F *:V 1CatV 2Cat F_\ast \;\colon\; V_1 Cat \longrightarrow V_2 Cat

between categories of V-enriched categories sending a V 1V_1-enriched category 𝒞\mathcal{C} to the V 2V_2-enriched category F *(𝒞)F_\ast(\mathcal{C}) such that

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

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

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

(Eilenberg-Kelly 65)

This operation F *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 𝒞\mathcal{C} has a single object and is hence (if thought of as pointed by that object) equivalently a monoid object in V 1V_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 1V 2V_1\rightleftarrows V_2 gives rise to a 2-adjunction V 1CatV 2CatV_1 Cat\rightleftarrows V_2 Cat (and also for profunctors).

  • VV-enriched categories can be defined more generally when VV is a multicategory, and any functor F:V 1V 2F: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 VV 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 1V_1 and V 2V_2 are cocomplete monoidal categories (or locally cocomplete bicategories), so that the bicategories V iV_i-Prof of V iV_i-categories and V iV_i-profunctors exist, change of enrichment also induces a lax functor V 1ProfV 2ProfV_1 Prof \to V_2 Prof. To make this version functorial on MonCat cocompleteMonCat_{cocomplete}, we need to consider V iProfV_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 VV we can construct another virtual double category VProfV Prof, and this construction is functorial.

Examples

Example

For any a monoidal category VV, the functor V(I,):VSetV(I,-): V \to Set is lax monoidal, hence induces a 2-functor from VCatV Cat to CatCat. This assigns to any VV-enriched category, 𝒞\mathcal{C}, its underlying ordinary category, usually denoted 𝒞 0\mathcal{C}_0, defined by 𝒞 0(x,y)=V(I,hom(x,y))\mathcal{C}_0(x,y) = V(I, hom(x,y)).

Example

If VV is a cocomplete monoidal category, the functor V(I,)V(I,-) above has a left adjoint that takes a set XX to the copower of XX copies of II. The resulting 2-functor CatVCatCat \to V Cat takes an ordinary category to the “free” VV-category it generates. Such VV-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 CatVCatCat \rightleftarrows V Cat.

Example

More generally, if an adjunction F:V 1V 2:UF: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=TopV_1 = Top and V 2=GTopV_2 = G Top for some topological group GG, then the right adjoint V 2CatV 1CatV_2 Cat \to V_1 Cat takes the “underlying topologically enriched category” of a category enriched in GG-spaces (its morphisms are the fixed-point spaces of the original GG-space-enriched category; we think of these as the “GG-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:SSetTop:SingReal:SSet \rightleftarrows Top : Sing between simplicial sets and topological spaces induces another adjunction SSetCatTopCatSSet Cat \rightleftarrows Top Cat between simplicially enriched categories and topologically enriched categories.

Example

If V=V=SSet or Top, then the set of connected components π 0:VSet\pi_0:V\to Set is lax monoidal. The resulting change of enrichment functor takes a VV-category CC to its “naive homotopy category” hC(x,y)=π 0(C(x,y))h C(x,y) = \pi_0(C(x,y)) obtained by “identifying homotopic morphisms”.

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

Example

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

Example

The nerve functor N:CatSSetN:Cat \to SSet preserves products and has a left adjoint τ 1\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:SetSetF =P \;\colon\; Set \to Set the power set-functor, the change of base functor P *:CatCatP_\ast \;\colon\; Cat \to Cat sends plain categories to plain categories. For CC any category, the morphisms of C polyP*(C)C^{poly} \coloneqq P\ast(C) are called the poly-morphisms of CC in Mochizuki 12, section 0.

References

Last revised on February 11, 2019 at 23:06:53. See the history of this page for a list of all contributions to it.