categorical generalizations of familiar objects

A couple of days ago I've learned that you can define trace in a very abstract setting.

Namely, suppose $F\colon A\to B$ is a functor between two categories. Suppose $E,G\colon B\to A$ are two functors, that are adjoint to $F$, such that $E$ is left adjoint and $G$ is right adjoint. Suppose moreover that $\nu\colon G\to E$ is a natural transformation. Then having this data $\forall x,y\in A$ you can define the trace map $$Tr\colon Hom(F(x),F(y))\to Hom(x,y)$$ for any $f\in Hom(F(x),F(y))$ to be the composition $$x\to GF(x)\to EF(x)\to EF(y)\to y,$$ where the first map is given by the adjunction unit $id_A\to GF$, second map is given by $\nu$, the third map is given by $E(f)$ and the last map is given by the adjunction counit $EF\to id_A$.

If we take $A=B=Vect_k$, $F=-\otimes V$ and $G=E=-\otimes V^*$ for some finite dimension vector space $V$ over $k$, and take $x=y=k$, then this construction gives the usual trace of a linear map.

From the top of my head I remembered a couple of more examples.

For example, if $C$ is a category, the Bernstein center of $C$ is defined to be $Z(C)=End(id_C)$, the (commutative monoid) of endomorphism of the identity functor. If the category $C$ is additive, then $Z(C)$ is a commutative ring. If we take $C$ to be the category of modules over some ring $R$, then $Z(C)$ is isomorphic to the center of $R$.

Another one, you can talk about open, closed subfunctors on the category of commutative rings into $Sets$. If your functors are representable, then you get the usual definition of open or closed subschemes.

So my question is: What are other nice examples of such categorical generalizations that you now?

A nice example is the notion of matrix. The most general setting (that I know of!) in which we can usefully think about matrices is in a category with finite biproducts. Common examples include abelian categories like categories of modules but there are even more general examples than this. See this blog post for some examples and more details. One gets back the prototypical case of finite-dimensional vector spaces over a field $k$ by decomposing every such vector space as a biproduct of copies of $k$.

Here are a few examples:

Kahler differentials

Everything I say here can be found in the nLab page on Kahler differentials. Notice that for a commutative ring $A$, the category of $A$-modules is equivalent to the category $\operatorname{Ab}(\mathsf{cRing}/A)$ of abelian group objects in the category commutative rings over $A$. One associates to an $A$-module $E$ the "square-zero extension" $A\oplus E$ with multiplication $(a,x)(b,y)=(ab,a y+b x)$. There has a natural forgetful functor $\operatorname{Ab}(\mathsf{cRing}/A)\to \mathsf{cRing}/A$, and the formation of Kahler differentials forms a left-adjoint to this. That is, $$ \hom_{\mathsf{cRing}/A}(B,A\oplus E) = \operatorname{Der}_A(B,E) = \hom_{A\text{-}\mathsf{mod}}(\Omega_{B/A}^1,E) = \hom_{\operatorname{Ab}(\mathsf{cRing}/A)}(A\oplus \Omega_{B/A}^1,A\oplus E) $$

This can be generalized to an arbitrary category. Following nLab, we replace $\mathsf{cRing}$ with $\mathsf{AffSch}$, and define, for any category $\mathsf{C}$, the tangent category of $\mathsf{C}$ to be the category of pairs $(x,A)$, where $x$ is an object of $\mathsf{C}$ and $A$ is an abelian group object in $\operatorname{Ab}(\mathsf{C}^\circ/x)$. Here I write $\mathsf{C}^\circ$ for the opposite category of $\mathsf{C}$. There is a forgetful functor from $T\mathsf{C}$ to the "arrow category" $\mathsf{C}^\to=\operatorname{Nat}(\bullet\to\bullet,\mathsf{C})$ that sends $(x,A)$ to the structure morphism $x\to A$. If we write $F:(T\mathsf{C})^\circ\to \mathsf{C}^\to$ for this functor, then we can define $\Omega^1$ to be the left-adjoint (if it exists) to $F$, i.e. $$ \hom_{x\backslash \mathsf{C}}(A,y) = \hom_{\operatorname{Ab}(\mathsf{C}^\circ/x)}(\Omega_{y/x}^1,A) $$ for $(x,A)$ in $T\mathsf{C}$ and $x\to y$ in $\mathsf{C}$.

This definition of Kahler differentials is "correct" in many contexts - simplicial commutative rings, smooth rings, ...

Lie algebras

This perspective on Lie algebras originated (I think) in SGA 3, exposé II. Let $S$ be a scheme and consider $\mathsf{Sch}_S$, the category of schemes over $S$. One starts by defining a functor from $S_\mathsf{qc}$, the category of quasi-coherent $\mathcal{O}_S$-modules, to schemes over $S$ by sending $\mathcal{M}$ to $I_S(\mathcal{M})=\operatorname{Spec}(\mathcal{O}_S\oplus\mathcal{M})$. (Astute readers will notice that this is exactly the previous construction). Now let $X$ be a presheaf on $\mathsf{Sch}_S$. One define, for each $\mathcal{M}$ in $S_\mathsf{qc}$, the tangent bundle of $X$ relative to $\mathcal{M}$ to be the functor $$ T_{X/S}(\mathcal{M}) = \underline{\hom}_S(I_S(\mathcal{M}),X):Y\mapsto \hom_S(I_S(\mathcal{M})\times Y,X) $$ (See the section on exponentials for an explanation of $\underline{\hom}$). If $X$ is represented by an actual scheme (which I'll call $X$), then $T_{X/S}=T_{X/S}(\mathcal{O}_S)$ is represented by $\mathbb{V}(\Omega_{X/S}^1)=\operatorname{Spec}(\operatorname{Sym}(\Omega_{X/S}^1))$.

If $X$ comes with an $S$-valued point (e.g. if $X=G$ is a group-valued functor with identity section $e:S\to X$) then the Lie algebra of $G$ is defined for any quasi-coherent $\mathcal{O}_S$-module $\mathcal{M}$: $$ \operatorname{Lie}(G,\mathcal{M}) \subset T_{G/S}(\mathcal{M}) $$

where $\operatorname{Lie}(G,\mathcal{M})(Y)$ is the set of $f\in T_{G/S}(\mathcal{M})(Y)=\hom_S(I_S(\mathcal{M})\times Y,G)$ such that the composite $Y\to I_S(0)\times Y = S\times Y = Y\xrightarrow{f} G$ is $Y\to S\xrightarrow{e} G$. (Note that I'm writing $X\times Y$ for $X\times_S Y$.) In other words, $\operatorname{Lie}(G,\mathcal{M})$ is the fiber product $S\times_G T_{G/S}(\mathcal{M})$. (In case it isn't clear, I'm being a little excessively loose in identifying schemes with their functor of points.)

If we let $\operatorname{Lie}(G) = \operatorname{Lie}(G,\mathcal{O}_S)$, then there is a general "adjoint action" $\operatorname{ad}:G\to \underline{\operatorname{Aut}}(\operatorname{Lie}(G))$, where $\underline{\operatorname{Aut}}$ is defined much like $\underline{\hom}$.

Subobject classifiers

This is much less complicated, and can be found in Mac Lane and Moerdijk's book Sheaves in Geometry and Logic. For $\mathsf{C}$ a category, a subobject of $x$ is an equivalence class of monomorphisms $u\to x$, where $u\to x$ and $v\to x$ are equivalent if they both factor through each other. One says that $\mathsf{C}$ has a subobject classifier if the functor that sends $x$ to $\operatorname{Sub}_\mathsf{C}(x)$, the class of subobjects of $x$, is represented by some $\Omega$ in $\mathsf{C}$. (In particular, if $\mathsf{C}$ has a subobject classifier, each object has a set, not a proper class, of subobjects.)

For $\mathsf{Set}$, the subobject classifier is $\Omega=\{0,1\}$, and the representability of $\operatorname{Sub}_\mathsf{C}$ is essentially the fact that subsets can be identified with their characteristic functions. However, more interesting categories (like sheaves on a site) also have subobject classifiers.

Exponentials

Further details here can also be found in Mac Lane and Moerdijk. One says that a category $\mathsf{C}$ with products has exponentials if for all $x$ in $\mathsf{C}$, the functor $y\mapsto x\times y$ has a left-adjoint. One denotes this adjoint by $(-)^x$, i.e. $$ \hom(x\times y,z)=\hom(y,z^x) $$ For $\mathsf{Set}$, $x^y$ is the set of functions from $y$ to $x$, and the adjunction expresses the fact that functions $f:y\times x\to z$ can be identified with their transpose: $y\mapsto (x\mapsto f(x,y))$. Once again, more interesting categories (like sheaves on a site) also have exponentials. In fact, if $F$ and $G$ are sheaves on a site $\mathcal{C}$, then the sheaf exponential $F^G$ is defined by $$ (F^G)(c)=\hom(G\times c,F) $$ where once again I identify $c$ with $\hom(-,c)$. Note that $F^G$ is sometimes written $\underline{\hom}(F,G)$. It's definition is not ad hoc either. One has (using the Yoneda lemma and the adjunction between exponentials and cartesian product) $$ (F^G)(c) = \hom(c,F^G)=\hom(G\times c,F) $$

Perhaps one of the most obvious examples is that of the nerve of a category (and so also the classifying space of a category.

The nerve $N(C)$ of a category $C$ is a simplicial set which in some sense encodes the topological structure of a category. The $0$-simplicies of $N(C)$ are the objects of $C$ and the set of $k$-simplicies $N_k(C)$ of the nerve are the towers of composable arrows of length $k$.

The face maps $d_i\colon N_k(C)\rightarrow N_{k-1}(C)$ are given by formally composing the tower at the $i$th position (or forgetting the arrows at the ends if $i=0$ or $k$). So for instance if $$\alpha=(f_0,f_1,f_2)\in N_4(C)$$ then $d_0(\alpha)=(f_1,f_2)$ and $d_1(\alpha)=(f_1\circ f_0,f_2)$.

The degeneracy maps $s_i\colon N_k(C)\rightarrow N_{k+1}(C)$ are given by formally inserting an identity arrow before $f_{i}$ (or after $f_{k-2}$ in the case $i=k-1$).

The classifying space $BC$ of $C$ is then the geometric realisation of the nerve $N(C)$. This generalises the usual construction of the classifying space $BG$ of a group $G$ by forming the usual category $C_G$ with one object whose arrows are the elements of $G$ with composition given by group multiplication. We get $$BG\cong BC_G.$$

The concept of a nerve of a category also generalises the concept of a nerve of an open covering of a topological space. If $X$ is topological space with open cover $\mathcal{A}=\{U_{\lambda}\}$ we denote by $N(\mathcal{A})$ the nerve of the covering $\mathcal{A}$. We also form the category (in this case a poset) $C_{\mathcal{A}}$ whose objects are the $U_{\lambda}$ and an arrow $U_{\lambda}\rightarrow U_{\lambda'}$ exists if $U_{\lambda}$ is a subset of $U_{\lambda'}$. We have $$N(\mathcal{A})=N(C_{\mathcal{A}}).$$