symmetric monoidal (∞,1)-category of spectra
The purpose of this summary is to show that the category (a variety of algebras) is cocomplete, using the general adjoint functor theorem. For a different proof in the context of Lawvere theories, see here.
Let us first fix some notation and recall the required facts on varieties of algebras. We assume the foundations of Categories Work (ZFC plus one fixed universe). Let be a small signature consisting only of function symbols (this set is assumed to be equipped with an arity function ), let be the set of -terms (also called derived operators), and let be a set of identities.
Recall that a -algebra is just an -structure (that is, a pair where is a set and is a function which assigns to each operator of arity a function ) that is a model for for any possible interpretation of the variables, for all .
A morphism is a function that satisfies for all and all of arity . The category consists of all small -algebras with the above morphisms as arrows.
There is an obvious forgetful functor , and it can be verified directly that this functor creates small limits (in fact, this functor is monadic and hence creates all limits). Consequently is small complete.
Before proving cocompleteness, some preparations are required. Let us write for short. Let be a small category. To prove cocompleteness, we will use the general adjoint functor theorem to construct a left adjoint for the diagonal functor .
Since is locally small and small-complete and is continuous (as any diagonal functor), all that is required is a solution set for each .
Recall that given a functor and an object of , an arrow of is said to span when, for all monic , if factors through , then is an isomorphism.
In our current context, we will replace ”monic” in the above definition by a stricter notion: the underlying function in is injective.
(CWM, p. 127). In a category , suppose that every set of subobjects of an object has a pullback. Then if preserves all these pullbacks, every arrow factors as for some arrow which spans and some .
It can be verified that this lemma still holds (in our context) if, in the definition of subobject and spanning arrow, ”monic” is replaced by ”the underlying function is injective.
If a cone spans , then is generated by .
Let be the subalgebra of generated by , and let be the inclusion. Then we have the cone obtained by restricting the codomains of the , and splits as . So is an isomorphism, and therefore must be the identity.
We can now finally prove
is small cocomplete.
Let us fix some . Define
We note that if , then by Lemma 2, , where
for some that spans , and . Since is small and each is small, so is (in detail, may be written as , and the image of a function from a -small set into the universe is -small). Now, as and are small, so is , and therefore we see that the underlying sets of all members of have cardinal numbers smaller than that of some fixed small set .
It follows that there exists a small set including one element from each isomorphism class of (for example, since is closed under isomorphisms, each isomorphism class includes an algebra whose underlying set is a subset of ).
(the set of all arrows for some ). To see that is small, note that each is in particular a function , and the set of all such functions is small (as any set of functions between two small sets).
Clearly, if for all the set of subobjects is small-indexed, then it follows from Lemma 1 that is the required solution set (since is small complete and is continuous), and we are done . But two injections with codomain are equivalent iff they have the same image in , and hence there is a bijection between the subobjects of and the subalgebras of , as required (recall that we replaced ”monic” by ”injective as a function” in the definition of subobject).
First, the category of algebras of a monad on has coequalizers; see the proof of proposition 3.4 (page 278 of 303) of Toposes, Triples, and Theories by Barr and Wells. So we have only to prove the following.
The algebra category has coproducts.
Let be the underlying functor, with left adjoint . Given a family of -algebras , there are canonical coequalizers
and since is the coproduct in the category of algebras, i.e., since coproducts of free algebras exist, the coproduct of the is constructed as a coequalizer of the pair
obtained by summing over each of the parallel pairs.
reflects reflexive coequalizers (since the underlying functor reflects classes of colimits preserved by the monad). Since the parallel pairs in the proof of the preceding proposition are reflexive, we see by adapting that proof that has coproducts. It remains only to prove the following.
If a category has finite coproducts and reflexive coequalizers, then it has general coequalizers.
Given a parallel pair , there is a reflexive parallel pair
whose coequalizer is the coequalizer of the pair .
See also cocompleteness of categories of algebras.