With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Given an endomorphism $f:A\to A$ in a traced monoidal category $\mathscr{C}$ there is a natural notation of a trace $\tr(f):1\to 1$, where $1$ is the tensor unit. When $\mathscr{C}$ is a $\mathbb{C}$-linear category for which the tensor unit (e.g. a fusion categories), we can canonically identity $\tr(f)$ with a complex number. That is, the trace of $f$ is the unique scalar $\lambda$ such that
In the case that $f$ is a linear endomorphism of a finite dimensional vector space, we recover the usual notion of trace from linear algebra.
The idea of the trace operation is easily seen in string diagram notation: essentially one takes an endomorphism $f: A\xrightarrow{}A$, and “closes the loop”.
This definition makes sense in any braided monoidal category, but often in non-symmetric cases one wants instead a slightly modified version which requires the extra structure of a balancing.
The trace of the identity $1_a:a \to a$ is called the dimension or Euler characteristic of $a$.
$C = Vect_k$ with its standard monoidal structure (tensor product of vector spaces): in this case $tr(f)$ is the usual trace of a linear map. Indeed, suppose $f$ is an endomorphism on a finite dimensional $k$-vector space $V$. Then, the trace $tr(f) \colon k\to k$ is given by the following composition.
Here $\varepsilon$ is the standard evaluation-pairing. The map $\eta$ is obtained from the dual linear map of $\varepsilon$ by applying the following isomorphisms, see (Dold & Puppe 1978).
One can then compute that
where $\{e_1,\ldots,e_n\}$ is a linear basis of $V$ and $\{e_1^\ast,\ldots,e_n^\ast\}$ is the dual basis. Now, if $f$ is given in this basis by a matrix $(a_ij)$, then
Finally, $(tr(f))(\lambda)$ is obtained by applying the pairing on this tensor: $(tr(f))(\lambda)=\lambda\sum a_{ii}$.
$C = SuperVect = (Vect_{\mathbb{Z}_2}, \otimes, b)$, the category of $\mathbb{Z}_2$-graded vector spaces with the nontrivial symmetric braiding which is $-1$ on two odd graded vector spaces: in this case the above is the supertrace on supervectorspaces, $str(V) = tr(V_{even}) - tr(V_odd)$.
$C = Span(Top^{op})$: here the trace is the co-span co-trace which can be seen as describing the gluing of in/out boundaries of cobordisms.
$C = Span(Grpd)$: this reproduces the notion of trace of a linear map within the interpretation of spans of groupoids as linear maps in the context of groupoidification and geometric function theory, made explicit at span trace.
In the symmetric monoidal (infinity,1)-category of spectra, the trace on the identity on a suspension spectrum of a manifold $X$ is the Euler characteristic of $X$ (see there).
See trace of a category.
See trace in a bicategory?.
Let $V \in C$ be a dualizable object, and $W$ any object. From an endomorphism $f$ of $V \otimes W$, one can produce an endomorphism $tr_V(f)$ of $W$, by applying the duality data of $V$. In terms of string diagrams, this is “bending along” the strand representing $V$.
TO DO: Draw the diagram just described.
Suppose $V$, $W$ are finite-dimensional vector spaces over a field, with dimensions $m$ and $n$, respectively. For any space $A$ let $L(A)$ denote the space of linear operators on $A$. The partial trace over $W$, Tr$_{W}$, is a mapping
Let $e_{1}, \ldots, e_{m}$ and $f_{1}, \ldots, f_{n}$ be bases for $V$ and $W$ respectively. Then $T$ has a matrix representation $\{a_{k l,i j}\}$ where $1 \le k,i \le m$ and $1 \le l,j \le n$ relative to the basis of the space $V \otimes W$ given by $e_{k} \otimes f_{l}$. Consider the sum
for $k,i$ over $1, \ldots, m$. This gives the matrix $b_{k,i}$. The associated linear operator on $V$ is independent of the choice of bases and is defined as the partial trace.
Consider a quantum system, $\rho$, in the presence of an environment, $\rho_{env}$. Consider what is known in quantum information theory as the CNOT gate:
Suppose our system has the simple state ${|1\rangle}{\langle 1|}$ and the environment has the simple state ${|0\rangle}{\langle 0|}$. Then $\rho \otimes \rho_{env} = {|10\rangle}{\langle 10|}$. In the quantum operation formalism we have
where we inserted the normalization factor $\frac{1}{2}$.
The categorical notion of trace in a monoidal category is due to
national Conference on Geometric Topology (Warsaw, 1978), pages 81{102, Warsaw, 1980. PWN.
and
Surveys include
Peter Selinger, A survey of graphical languages for monoidal categories (pdf), Section 5
Kate Ponto, Mike Shulman, Traces in symmetric monoidal categories (pdf).
Generalization of this to indexed monoidal categories is in
and to bicategories in
Further developments are in
Andre Joyal, Ross Street, and Dominic Verity, Traced Monoidal Categories
David Ben-Zvi, David Nadler, Nonlinear traces (arXiv:1305.7175)
For the notion of partial trace, particularly its application to quantum mechanics, see:
Last revised on November 2, 2023 at 04:01:47. See the history of this page for a list of all contributions to it.