Maybe I’m misunderstanding your notation, but doesn’t that equality simply follow from the naturality of the universal property of the cartesian product?

]]>Actually, I’m not sure about the semantics of a similar example, where the generator can be inside or outside the pair:

$g_1:(B_1,B_2) \to C \qquad g_2:(B_1,B_2) \to D$

$[g_1(f_{(1)}(x),f_{(2)}(x)),g_2(f_{(1)}(x),f_{(2)}(x))]\;:\;C \& D$

Dispensing with tensor projections, this could be written in two ways:

$let\;(b_1,b_2) = f(x)\;in\;[g_1(b_1,b_2),g_2(b_1,b_2)]$

$[let\;(b_1,b_2) = f(x)\;in g_1(b_1,b_2),let\;(b_1,b_2) = f(x)\;in g_2(b_1,b_2)]$

Are these actually equal? Maybe sometimes implicit $\otimes$-elim is just plain wrong, not tricky.

]]>I added a remark about monoidal topology and bicategories of matrices to the section on Lawvere metric spaces, with a reference, and deleted the incomprehensible section.

]]>Clarified the difference between concrete and abstract clones (following wikipedia for terminology). I don’t have time to add any more, but now at least the idea and definition sections aren’t contradictory.

]]>For your entertainment, I’ll tell you about a tricky situation I ran into involving your tensor projections and generators, combined with $\&$-intro.

For both examples,

$x:A \qquad f:A \to (B_1,B_2)$

Example 1:

$g_1:B_1 \to C \qquad g_2:B_1 \to D \qquad h:B_2 \to E$

$([g_1(f_{(1)}(x)),g_2(f_{(1)}(x))],h(f_{(2)}(x)))\;:\;(C \& D) \otimes E$

Example 2:

$g:(B_1,B_2) \to C \qquad h:A \to D$

$[g(f_{(1)}(x),f_{(2)}(x)),h(x)]\;:\;C \& D$

The problem is whether you use the generator rule on $f$ inside or outside the $\&$-intro. In example 1, you have to use it outside; in example 2, you have to use it inside the left component. But processing both examples left to right, the $f$ is first encountered in the left component, because the $\otimes$-elim is implicit.

I’m not sure if I’m going to handle this.

I guess there’s basically the same problem with $\oplus$-elim.

]]>added pointer to

- Mathew Bullimore, Tudor Dimofte, Davide Gaiotto,
*The Coulomb Branch of 3d $\mathcal{N}=4$ Theories*, Commun. Math. Phys. (2017) 354: 671 (arXiv:1503.04817)

also this one:

- Mathew Bullimore, Tudor Dimofte, Davide Gaiotto,
*The Coulomb Branch of 3d $\mathcal{N}=4$ Theories*, Commun. Math. Phys. (2017) 354: 671 (arXiv:1503.04817)

for hyperlinking references

]]>for hyperlinking references

]]>added pointer to

- Benjamin Assel, Stefano Cremonesi, Section 2.1 of:
*The Infrared Physics of Bad Theories*, SciPost Phys. 3, 024 (2017) (arXiv1707.03403)

The fact that the system of Hilbert spaces acts as a repository for discrete dynamic geometric data and dynamic fields opens the possibility to convert the system into a self-creating model of physical reality in which the clock of the universe starts ticking after that the repository is filled with data that tell the complete life stories of the elementary particles. This possibility is explored in "A Self-creating Model of Physical Reality"; http://vixra.org/abs/1908.0223 and is presented in http://www.e-physics.eu/Base%20model.pptx ]]>

starting something – for the moment just to record this reference:

- Washington Taylor,
*Adhering 0-branes to 6-branes and 8-branes*, Nucl. Phys. B508: 122-132, 1997 (arXiv:hep-th/9705116)

brief `category:people`

-entry for hyperlinking references at *configuration space of points* and at *correlator as differential form on configuration space of points*

finally added pointer to

- Christopher Beem, David Ben-Zvi, Mathew Bullimore, Tudor Dimofte, Andrew Neitzke,
*Secondary products in supersymmetric field theory*(arXiv:1809.00009)

Well, in principle both would be nice to have – one for simplicity of exposition, the other for convenience of practical use. I would say start with whichever is easier for you.

]]>added further and improved illustration of the trace operation sending horizontal chord diagrams to round chord diagrams (here)

]]>To handle lollipop (internal-hom), do you want unidirectional inference with annotations on lambdas, or bidirectional typing with an optional annotation/cut rule. Do you want to require eta-long terms?

]]>Added doi links for the references.

]]>added pointer to

- Rafe Mazzeo, Edward Witten,
*The Nahm Pole Boundary Condition*, In:*The influence of Solomon Lefschetz in geometry and topology*, Contemporary Mathematics 621 (2014): 171 (doi:10.1090/conm/621)

Only a subtle difference exists between a vector space and a Hilbert space. In this way it becomes possible that a huge number of separable Hilbert spaces can share the same underlying vector space. Quaternionic number systems exist in many versions that distinguish between the Cartesian and polar coordinate systems that sequence their members. This affects the symmetry of the number system. A Hilbert space selects a version of the number system for specifying its inner product. This selects the symmetry of that Hilbert space. Each separable Hilbert space can manage a private parameter space in the eigenspace of a dedicated normal operator (that I call reference operator) by letting that eigenspace represent by the rational values in the selected version of the number system that is used to specify the inner products of vector pairs. A special category of normal operators can be defined by letting them share the eigenvectors of the reference operators and replacing the corresponding eigenvalues of the reference operator by the target values of a selected function. Each infinite dimensional separable Hilbert space owns in this way a unique non-separable Hilbert space that embeds its separable companion. In this way the special category of normal operators become field operators that combine Hilbert space operator technology with function theory, differential calculus and integral calculus. The continuum eigenspaces of these operators in the non-separable Hilbert space will implement a general field theory that in case of a quaternionic number system treats dynamic fields in a well-defined way.

In this way a system of Hilbert spaces can act as a structured repository for discrete dynamic geometric data and dynamic continuums that act like (physical) fields. ]]>

starting something – just a bare minimum for the moment

]]>some minimum, for the moment just so as to bring in infrastructure for stating the *Jacobson-Morozov theorem*

added pointer to

- Davide Gaiotto, Edward Witten,
*Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory*, J Stat Phys (2009) 135: 789 (arXiv:0804.2902)

added these pointers:

An interpretation of the s-rule for D-brane intersections with NS5-branes (Dp-D(p+2) brane intersections and Dp-D(p+4) brane intersections) as a version of the Pauli exclusion principle is discussed in:

Constantin Bachas, Michael Green, Adam Schwimmer, Section 2.3 of:

*$(8,0)$ Quantum mechanics and symmetry enhancement in type I’ superstrings*, JHEP 9801 :006, 1998 (arXiv:hep-th/9712086)Constantin Bachas, Michael Green,

*A Classical Manifestation of the Pauli Exclusion Principle*, JHEP 9801 (1998) 015 (arXiv:hep-th/9712187)