When forgetting structure doesn't matter

Question

What forgetful functors are equivalences?

The motivation here is understanding when some part of a structure can be 'safely' forgotten, even if remembering it might make our lives easier.

There is some ambiguity as to what constitutes 'a forgetful functor', but I am open to any answers that are 'obvious' forgetful functors or where the interpretation is justified.

I don't have any nontrivial examples (the forgetful functor from the $EM$-category of the trivial monad on a category), and I am open to arguments that no nontrivial examples exist. Any relevant input is appreciated.

Answer 1

The functor that takes a simplicial abelian group to its associated chain complex is arguably a "forgetful" functor. By the Dold-Kan theorem it induces an equivalence of categories.

Answer 2

By the positive solution to Hilbert's fifth problem (as well as the theorem of Cartan that continuous homomorphisms between Lie groups are automatically smooth), the forgetful functor from the category of Lie groups (with smooth homomorphisms as the morphisms) to the category of locally Euclidean topological groups (with continuous homomorphisms as the morphisms) is an equivalence.

I would imagine that many rigidity theorems (e.g., Mostow rigidity) can also be reformulated as an assertion that something resembling a forgetful functor is in fact an equivalence, but I have not worked this out carefully. (For instance, Gromov's quasi-isometric rigidity program can be sort of interpreted as the question of measuring the extent to which the forgetful functor from finitely generated groups (with virtual homomorphisms rather than global homomorphisms as morphisms) to metric spaces (with quasi-isometries as the isomorphisms) fails to be an equivalence. The Ribe program to analyze the local geometry of Banach spaces via their metric structure (up to uniformly bicontinuous maps) is in a similar spirit, and inspired by the Ribe rigidity theorem, though a little less easy to describe explicitly in terms of forgetful functors.)

Answer 3

The forgetful functor from group objects in abelian groups to abelian groups (just forget the extra group structure) is an equivalence.

More generally there are lots of "idempotent" constructions, such as "taking abelian group objects in" or "taking pointed objects in" that give examples like this. These are the "uninteresting" ones in that they are very general.

An example not of this type is given by duality data. Let's say you have symmetric monoidal category $C$. Inside there you have dualizable objects, and they form a full subgroupoid $C^{dbl}$. But a dualizable object can also be seen as a bunch of duality data: a tuple $(x,y, x\otimes y \to 1)$ for example. This type of data forms a groupoid $DDat(C)$ and the forgetful functor $DDat(C) \to C^{dbl}$ is an equivalence.

Another example that happens a lot in ordinary category theory is when forgetting some higher dimensional coherence data ends up not being necessary. For example, forgetting from "very structured monoids" to simply monoids is an equivalence, where a "very structured monoid" is a functor $\Delta^{op}\to C$ satisfying the Segal conditions - similarly with "very structured commutative monoids", or more generally algebras or commutative algebras.

Similarly one can forget from very structured monoidal categories to monoidal categories - although that is an equivalence only if one views both as $(2,1)$-categories. But I think the above shows that this phenomenon occurs even in ordinary categorical situations.

Let me give a final example: there are lots of criteria like "to preserve colimits, it suffices to preserve coequalizers and coproducts". This can be seen as a forgetful equivalence: from the category whose objects are cocomplete categories and arrows are colimit preserving functors to the category with the same objects but functors are functors which preserve only certain types of colimits. These statements are easy, but it is still an example of this form.

Answer 4

My favorite such functor is the forgetful functor from "$C^\infty$ manifolds with real coefficients" (defined as spaces with a sheaf of $\mathbb{R}$-algebras locally isomorphic to the sheaf $C^\infty_\mathbb{R}$ on the space $\mathbb{R}^n$) to $C^\infty$ manifolds with complex coefficients (define as spaces with a sheaf of $\mathbb{C}$-algebras locally isomorphic to the sheaf $C^\infty_{\mathbb{C}}$ on $\mathbb{R}^n$). This equivalence fails to hold in the supergeometric world (as well as in various other extensions of the theory of real manifolds).

Answer 5

The forgetful functor from abelian varieties to pointed schemes is fully faithful, (hence an equivalence onto its essential image). That is, a pointed map of schemes between abelian varieties automatically preserves the multiplication.

(Maybe some assumptions about the base need to be made for this to be true? I'm not sure. At any rate, I'm pretty sure I've been told that this is at least true over $\mathbb C$.)

Answer 6

An example from a pedagogical perspective: the category of based vector spaces, i.e. vector spaces with a chosen basis, and morphisms matrices, is equivalent to the category of vector spaces by the functor which forgets the basis. Similarly: the category of homotopy types with a choice of CW complexes realizing the type, and morphisms homotopy classes of continuous maps. Generally, this happens any time a choice of extra structure is always possible and doesn’t change the morphisms, which of course is just rephrasing the request that the forgetful functor be essentially surjective and fully faithful!

Answer 7

One reason to believe that this phenomenon is ubiquitous yet not so interesting is the following construction:

Given an equivalence of categories $F \colon \mathscr C \stackrel\sim\to \mathscr D$, define the category of triples $(c,d,f)$ of an object $c \in \mathscr C$, an object $d \in \mathscr D$, and an isomorphism $f \colon F(c) \stackrel\sim\to d$. It has forgetful projections to $\mathscr C$ and $\mathscr D$ that are both equivalences.

For an example of this phenomenon in action, consider the equivalence between Boolean rings and Boolean lattices. It's not really forgetful in the sense that you need to define some new structure going both ways. However, we can define a category of Boolean ring lattices $(A,+,\cdot,0,1,\wedge,\vee)$ expressing both sets of axioms and the relations between them (ok, in this case $\cdot$ and $\wedge$ are literally the same operation). Then both forgetful functors are equivalences.

Likewise, there is an equivalence between algebraic lattices $(L,\wedge,\vee)$ and posets $(P,\leq)$ with finite products and coproducts, and one can consider the object $(L,\wedge,\vee,\leq)$ remembering both. Then the forgetful functors are equivalences.

(In both examples, we only work with $f \colon F(c) \stackrel\sim\to d$ that are the identity on some fixed object $F(c) = d$.)

Answer 8

Group objects in the category of smooth manifolds (aka Lie groups) are the same thing as monoid objects in the category of smooth manifolds whose underlying monoid is a group.

Said in other words: the inverse map, when it exists, is automatically (continuous and) smooth.

Answer 9

An important example from type theory/categorical logic: the forgetful functor from categories with families (or variants) to categories with attributes (or variants).

Briefly, both are category-based structures encoding the structure of a dependent type theory (and hence the structure needed to interpret such a theory). A category with families consists of a category of contexts, a presheaf of types over contexts, and a further presheaf of terms indexed over types, together with an operation of context extension satisfying a certain universal property.

The universal property forces terms to correspond to sections of certain projection maps. Categories with attributes take advantage of that, including just the category of contexts, the presheaf of types, the extension operation, and the projection maps.

So the forgetful functor is indeed an equivalence, and genuinely not an isomorphism: Given a CwF, if you forget to the underlying CwA, you can recover original the presheaf of terms just up to isomorphism.

Each notion has some advantages. CwA’s are a little “tighter” and easier to set up, since they involve less data; but taking the preheaf of terms as primary, in CwF’s, simplifies some later constructions, and allows very fruitful analyses and generalisations.

Answer 10

There is an obstruction to nontrivial examples. With strong enough metatheory (with enough global choice), equivalent categories of structures (in which the isomorphism class of an object is a proper class) are isomorphic. I think the argument is not very standard, so I will summarize it here.

Let $F$ be an equivalence between $C$ and $D$. For each isomorphism class of objects in $C$, fix a bijection from that isomorphism class to the isomorphism class containing its direct image under $F$. Define $G$ from the objects of $C$ to the objects of $D$ as follows:

For each object $A$, let $G(A)$ be the image of $A$ under the fixed bijection $g_{[A]}$ from the isomorphism class of $[A]$ to the isomorphism class $[F(A)]$.

For each object $A$, choose an isomorphism $h_A$ from $F(A)$ to $G(A)$.

For each arrow $s$ from $A$ to $B$, let $G(s)$ be the arrow $h_B.F(s).(h_A)^{-1}$ from $G(A)$ to $G(B)$.

Since $F$ is an equivalence, one can check that $G$ is an isomorphism of categories.

Therefore, if a forgetful functor (between categories of structures) is an equivalence, then the categories must be isomorphic.

Answer 11

Let $\mathcal{F}$ be the forgetful functor from the category of compact uniform spaces (the compact uniform spaces are the complete and totally bounded uniform spaces) to the category of compact proximity spaces. Let $\mathcal{G}$ be the forgetful functor from the category of compact proximity spaces to the category of compact Hausdorff spaces. Then the functors $\mathcal{F},\mathcal{ G},\mathcal{G}\circ\mathcal{F}$ are isomorphisms, so $\mathcal{F},\mathcal{G},\mathcal{G}\circ\mathcal{F}$ are equivalences of categories as well.

More generally, if $(X,\mathcal{U}),(Y,\mathcal{V})$ are uniform spaces, $(X,\mathcal{U})$ is compact, and $f:X\rightarrow Y$ is continuous, then $f$ is automatically uniformly continuous.

Answer 12

What about the classical theorem that the natural transformation $\eta$ witnessing the adjointess of two functions $F, G$:

$$ \alpha_{X, Y} : Hom(FX, Y) \simeq Hom(X, GY) $$

is determined by the values $$\epsilon_X:= \alpha_{X, FX} (1_{FX}) : X \to GFX$$ $$\eta_Y:= \alpha_{GY, Y}^{-1}(1_{GY}) : FGY \to Y$$

I always found this argument clever and simple, despite it is very basic.

Answer 13

The forgetful functor from Z-modules to abelian groups (which forgets the scalar multiplication) is an isomorphism.

Answer 14

$\mathrm{C}^*$-algebras $\to$ Banach algebras that happen to admit an involution satisfying the $\mathrm{C}^*$-algebra axioms.

Answer 15

For $k$ big enough, any $C^k$ map between finite-dimensional vector spaces which commutes with scalar multiplication is a homomorphism.