Recall that a presentable (∞,1)-category is a localization of a (∞,1)-category of (∞,1)-presheaves. In particular it has all small colimits. An (∞,1)-functor from the cartesian product of two presentable -categories is bilinear if it respects colimits in both variables.
It turns out that there is a universal such bilinear functor
The collection of presentable -cateories with colimit-preserving (∞,1)-functors between them (i.e. with “linear” functors between them!), is an -generalization of the category of ordinary categories and bimodules or profunctors, or distributors between them. See distributor and in particular the discussion there about the equivalent reformulation in terms of colimit-preserving functors.
Using with its notion of “linearity” one obtains a very general notion of -linear algebra. This is described at geometric ∞-function theory.
The symmetric monoidal structure on presentable -categories restricts to one on presentable stable (∞,1)-categories.
The tensor unit of stable presentable -categories is the stable (∞,1)-category of spectra.
A functor between locally presentable -categories:
For , let
Evidently would be the hom--category of an enhancement of to an -category; its maximal sub--groupoid is the hom--groupoid .
For all we have that is itself locally presentable.
This is HTT, prop. 126.96.36.199.
This is HTT, prop. 188.8.131.52.
Let be a finite collection of locally presentable -categories. There exists a locally presentable -category and an (∞,1)-functor
(the tensor product) such that
it preserves (∞,1)-colimits in each variable;
for every , composition with produces an equivalence of (∞,1)-categories
onto the full sub-(∞,1)-category of those functors, that preserves colimits in each argument.
This is (Lurie, NA, theorem 4.1.4).
This tensor product makes a symmetric monoidal (∞,1)-category. Indeed it is even closed, since the hom-objects supply adjoints to the tensor product.
The definition of the tensor product is easy: , where denotes the category of continuous (small-limit-preserving) functors. Equivalently, this is , where denotes the category of cocontinuous (small-colimit-preserving) functors; it is tempting to write this as , but note that is not itself locally presentable. However, is nevertheless cocomplete, so any cocontinuous functor has a right adjoint.
To show that is locally presentable, recall that for some small -cocomplete . Thus , where denotes the category of -limit-preserving functors; this is an accessible localization of and hence locally presentable.
To show that has the right universal property, we compute
To justify the step , note that both sides are equivalent to some subcategory of functors , and in both cases part of the requirement is that -limits be preserved in the -variable and all small limits be preserved in the -variable. The remaining condition on the left is that for each the corresponding functor is accessible, i.e. preserves -filtered colimits for some cardinal . On the right, the remaining condition is that the curried functor is accessible, i.e. preserves -filtered colimits for some cardinal . Since -filtered colimits commute with -limits as long as , such colimits in are pointwise, so we are at least talking about the same colimits, so the right-hand condition implies the left (take each ). On the other hand, because is small, if the left-hand condition holds we can take to make the right-hand condition hold.
Above we remarked that the forgetful functor preserves limits. It does not preserve colimits, and while has colimits they are in general not explicit. However, there are some cases in which they can be described explicitly, such as when they coincide with limits. For instance:
Small coproducts in coincide with small products.
Let be a product in . Then we have
To be precise, the next result requires an enhancement of to an -category so that we can talk about powers and copowers by small -categories. However, the proof gives an explicit universal property that makes sense without having to define that whole -category.
The copower of by a small -category is .
Let denote the presheaf (∞,1)-category? on . This is its free cocompletion, so we have . Dually, is the free completion of , so we have . Now we have:
In general, this sort of argument should work for all lax colimits of lax functors; for instance, Kleisli objects should also coincide with Eilenberg-Moore objects? (though this also requires enhancing to an -category). The corresponding 1-categorical fact is that in any bicategory with local colimits? (colimits in each hom-category distributed over by composition), lax colimits of lax functors are also lax limits.
In some context it makes good sense to think of as a model for an -category of “-vector spaces”, or at least “-abelian groups”. For instance, the fact that the -category has local colimits, making certain limits and colimits coincide, is a sort of categorification of the fact that of Vect and Ab are additive categories in which finite products and coproducts coincide.
More on this analogy is at integral transforms on sheaves. Here a small -category is to be thought of as a basis and the locally presentable -category as the -vector space spanned by this basis. The colimits in play the role of addition of vectors and the fact that morphisms in are colimit-presserving means that they play the role of linear maps between vector spaces. This is described also at Lawvere distribution.
The monoidal product plays the role of the tensor product of vector spaces, with a morphism out of being a bilinear morphism out of , and the fact that is closed monoidal reflects the fact that Vect is closed monoidal. The construction of the tensor product as corresponds to the fact that, at least for finite-dimensional , we have .
Combined with the fact that the embedding preserves limits (prop. 2), this yields some useful statements.
For instance with regarded as , for any ∞-group with delooping ∞-groupoid , we may think of an (∞,1)-functor as a linear representation of : the single object of is sent to a presentable -category and the morphisms in then define an action of on that.
The -category is introduced in section 5.5.3 of
The monoidal structure on is described in section 4.1 of
That this is in fact a symmetric monoidal structure is discussed in section 6 of
(see proposition 6.14 and remark 6.18). This monoidal structure is exhibited as a restriction of a monoidal structure on all cocomplete -categories in section 4.8 of