Hussain Kadhem
2022-09-06
Consider a category \(C\). It can be thought of as something that models the structure of the following diagram: \[\begin{matrix} &\swarrow & \leftarrow & \nwarrow_S&\\ C_0 && {}^I\rightarrow && C_1\\ &\nwarrow & \leftarrow & \swarrow^T& \end{matrix}\]
Here \(C_0\) is the collection of objects, \(C_1\) is the collection of morphisms, \(S\) and \(T\) are the source and target assigning functions, and \(I\) is the identity assigning function.
As well as composition laws, expressed as completions of all triangles \[\begin{matrix} &&c'&&\\ &\nearrow_f && \searrow^g\\ c && \dashrightarrow^h && c'' \end{matrix}\]
We can think of \(h\) as the composition of \(g\circ f\), but we can also think of the whole triangle as a kind of geometric witness of the compositional structure.
Let us introduce a new collection, \(C_2\), containing diagrams \(t\) of the form: \[\begin{matrix} &&t_1&&\\ &\nearrow_f & \Downarrow^t & \searrow^g\\ t_0 && \rightarrow^h && t_2 \end{matrix}\] We equip \(C_2\) with edge-assigning functions \(d_i \mid 0 \leq i \leq 2\), sending a triangle to the edge opposite the vertex \(t_i\). For any arrow \(c \to^f c'\), we can also send it to the degenerate triangle \(I_0(f)\) given by \[\begin{matrix} &&c'&&\\ &\nearrow_f & \Downarrow^{I(f)} & \searrow^1\\ c && \rightarrow^f && c' \end{matrix}\] These send an edge to the degenerate triangle where the edge opposite vertex \(t_i\) is the identity.
For consistency, let us rename the functions \(S,T\) to \(d_0,d_1\). We can then fit this new collection of triangles into an augmented diagram which is also modeled by the category:
\[C_0 \rightleftarrows_d^I C_1 \rightleftarrows_d^I c_2.\]
The upshot is that this single diagram also captures some of the compositional structure. To get the rest of it, we need to impose a sort of lifting property on partial triangles. Suppose that \(c \to c' \to c''\) fit into an incomplete triangle like the one from before. Then we require the existance of a unique triangle \(t\) which `fills it in.’
Exercise. Notice that we do not require this completion for partial diagrams which are missing one of the diagonal arrows. Show that this constrains all morphisms to be invertable.
It is this geometric structure of ordinary categories which we will generalize to define \(\infty\)-categories. Whereas the former are models of triangular structures with edge-completion constraints, the latter will be models of higher-dimensional simplices with face-completion constraints.
Denote by \([n]\) the ordered sequence \((0 \to 1 \to \cdots \to n)\). These are abstractions of the standard \(n\)-simplices, where we consider only subsets of their vertices and the combinatorics thereof. The simplex category \(\Delta\) has as objects \([n]\), and morphisms generated by the following order-preserving functions:
\(d_i[n]\) for \(0 \leq i \leq n\): the function \([n] \to [n+1]\) which is injective and skips \(i\).
\(I_i[n]\) for \(0 \leq i < n\): the function \([n] \to [n-1]\) which is surjective and equal on \(i,i+1\).
The functions \(d\) are face-assigning functions, and the functions \(I\) are degeneracy-assigning functions.
A simplicial set is a presheaf on \(\Delta\). That is, for each nonnegative integer \(n\) it assigns a set of \(n\)-simplices, which are compatible with face and degeneracy pullbacks.
For some category \(C\), a simplicial object in \(C\) is a functor \(\Delta^{\operatorname{op}} \to C\).
Let \(S_\bullet\) be a simplicial object. The horn \(\Lambda^k[n]\) of \(S\) is the union of all but the \(k\)th faces of \(S_n\). We call the horn outer if \(k=0\) or \(k=n\), and otherwise inner. The intuition is that inner horns should generalize the partial triangles from earlier.
\(S\) is a weak Kan Complex if it satisfies the following lifting property: every inner horn \(\Lambda^k[n]\) lifts to a simplex \(\Delta[n]\) which completes it. The intuition is that the simplex corresponds to a way of composing the faces of the horn, and the missing face is the result of such a composition.
CAUTION Notice that we did not require the lifts to be unique! This is a weakening of the constraint we required earlier for ordinary categories. In general, we only require that any two lifts are in a relationship whitnessed by a higher simplex, and usually that it is invertable.
A \(\infty\)-category is modeled by a simplicial set \(C_\bullet\) which is a weak Kan complex. We think of the simplices \(C_n\) as being higher dimensional morphisms, with composition laws given by the weak Kan lifts.
There are two important classes of examples of these structures:
For any topological space \(X\), we can associate to it the singular chain complex \(\operatorname{sing}_\bullet(X)\), with \(n\)-simplices given by continuous maps into \(X\) from the standard \(n\)-simplex, and faces and degeneracies given by the standard maps.
For any category \(C\), we can associate to it the nerve \(N_\bullet(C)\). It has as \(n\)-simplices diagrams of the form \(c_0 \to^{f_1} c_1 \to \cdots \to^{f_n} c_n\), face maps \(d_i\) given by deleting the object \(c_i\), and degeneracies \(I_i\) given by repeating the object \(c_i\).
Exercise. Show that these simplicial sets satisfy the weak Kan lifting property. (The second is straightforward, and the first follows from a retraction argument.)
A monoidal category is a category \(C\) equipped with a tensor product functor \(\otimes: C \times C \to C\), a unit object \(1_C\), and natural transformations for associativity and left/right unity. These are required to fit into compatibility conditions given by a triangle identity for the unitors, and a pentagon identity for the associator. (Rumour has it that if you write these out explicitly for an actual example, then the category theory devil will suddenly inhabit your space, make all your transformations unnatural, and force you to make non-canonical choices.)
A category \(C\) enriched in a monoidal category \(V\) is like an ordinary category, but the hom sets \(C(c,c')\) are instead given by an object in \(V\). Composition is expressed by requiring a morphism from \(C(c,c')\otimes_V C(c',c'') \to C(c,c'')\).
A natural example is the category of vector spaces and linear maps, which is enriched in itself.
A category with one object which is enriched in abelian groups is a ring.
Let \(R\) be a commutative ring. A differential graded (dg) category is a category enriched in chain complexes of \(R\)-modules.
A dg-category with one object is a differential graded algebra. (An example of this is the algebra of differential forms.)
The archetypal example of a dg category is the category of chain complexes of \(R\)-modules, with mapping complexes as hom objects.
DG-categories are secretly \(\infty\) categories. One way to witness this is using the dg nerve construction, in which the \(n\)-morphisms are maps in some mapping complex of degree \(n-1\), and composition is witnessed by a cycle condition.
A filtered category generalizes the idea of a directed set. Whereas the latter are used to define nets in topology, the former will be used to define certain kinds of limiting processes of categories.
A filtered category \(C\) is one that satisfies the following conditions:
For every two objects \(c',c''\), there exists an object and morphisms \(c \to c'\), \(c \to c''\).
For any two parallel morphisms \(c \to^f_g c'\), there exists an arrow \(c' \to^h c''\) equalizing them (that is, such that \(h \circ f = h \circ g\).)
A cofiltered category is one for which its opposite category is filtered.
A filtered colimit is the colimit of a diagram from a filtered category, and a cofiltered limit is the limit of a diagram from a cofiltered category.
Filtered colimits are classified by the key property that they commute with finite limits.
Let \(C\) be a category. In general, it might not be that \(C\) contains all colimits. However there is a vary natural way of formally adjoining all colimits in \(C\).
This is given by the Yoneda embedding, which includes \(C\) into its category of presheaves. \(\operatorname{psh}(C)\) is the free cocompletion of \(C\), as it contains all formal colimits of \(C\), and is universal among categories that do so.
Now we can define an ind object in \(C\). It is a filtered colimit in the free cocompletion of \(C\). That is, it is a formal filtered colimit in \(C\), which in general is only witnessed by a presheaf on \(C\).
A site \((C,J)\) is a category \(C\) equipped with a structure \(J\) called a coverage.
A coverage is the assignment, to each object \(U\) in \(C\), a collection of covering families \(\{U_i \to^{f_i} U\}\), which are stable under pullback to a different object \(U'\). (The stability condition can be stated in various ways which we don’t care about.)
A sheaf on a (nice) site \((C,J)\) is a presheaf \(X\) on \(C\), such that the following diagram is an equalizer: \[X(U) \to \prod_i X(U_i) \rightrightarrows \prod_{j,k} X(U_j\times_U U_k).\]
There are in fact other niceness conditions which can be imposed, but we won’t go into these.
Consider the category of affine schemes, which is opposite to the category of commutative rings. We can turn it into a site by equipping it with a coverage structure coming from the Zariski topology.
One way to think of a scheme \(X\) is as a sheaf on this site. It sends each ring \(R\) to the set of \(R\) points in \(X\).
(There are some more technical compatibility conditions which we won’t go into.)
A simplicial commutative ring is a simplicial object in the category of commutative rings. Similar to how affine schemes are categorically equivalent to commutative rings, derived affine schemes are equivalent to simplicial commutative rings. In characteristic \(0\), the latter is equivalent to the theory of commutative dg algebras.
A derived scheme will be a generalization of the functor-of-points approach to the setting of derived affine schemes. There will be some technical detail to be covered later, but the right way to think about derived schemes is as some sort of `locally presentable’ \(\infty\)-sheaf on the \(\infty\)-site of simplicial commutative rings.