4845:
6740:
5295:
5151:
5139:
5127:
6987:
7007:
6997:
6352:
6041:
5998:
5958:
5933:
5910:
5891:
5872:
5841:
5697:
4677:
5281:
4434:
Note that because the diagram is in generic form (i.e. each layer contains exactly one box) the definition of tensor is necessarily biased: the diagram on the left hand-side comes above the one on the right-hand side. One could have chosen the opposite definition
2286:. Intuitively, the edges in a progressive plane graph go from top to bottom without bending backward. In that case, each edge can be given a top-to-bottom orientation with designated nodes as source and target. One can then define the domain and codomain
3406:
4428:
4526:
4542:
3275:. One such definition is to define string diagrams as equivalence classes of well-typed formulae generated by the signature, identity, composition and tensor. In practice, it is more convenient to encode string diagrams as formulae in
2132:
4679:
That is, if the boxes in two consecutive layers are not connected then their order can be swapped. Intuitively, if there is no communication between two parallel processes then the order in which they happen is irrelevant.
909:
3529:
2660:
4172:
2351:
1194:
1132:
3735:
2501:
2198:
1996:
4030:
1383:
5043:
2452:
5003:
3861:
1303:
5466:
Many extensions of string diagrams have been introduced to represent arrows in monoidal categories with extra structure, forming a hierarchy of graphical languages which is classified in
Selinger's
5169:
3614:
2740:
1070:
1014:
4857:
A monoidal category is equivalent to a 2-category with a single 0-cell. Intuitively, going from monoidal categories to 2-categories amounts to adding colours to the background of string diagrams.
534:
1831:
5355:
5174:
3793:
5419:
1655:
2581:
4329:
4281:
4822:
5116:
1717:
958:
472:
3238:
3054:
4915:
3660:
5420:
5081:
390:
3161:
1938:
1872:
1546:
1587:
1231:
803:
2853:
2284:
2029:
3309:
642:
4963:
4939:
4233:
3200:
3115:
2056:
1782:
1751:
1487:
1437:
837:
767:
4198:
3920:
686:
4763:
601:
569:
425:
321:
210:
4836:
The parallel composition of 2-cells corresponds to the horizontal juxtaposition of diagrams and the sequential composition to the vertical juxtaposition of diagrams.
3300:
2528:
1678:
1460:
1407:
736:
3887:
2985:
2888:
2779:
1898:
3023:
2946:
2817:
2251:
5163:
The string corresponding to the identity functor is drawn as a dotted line and can be omitted. The definition of an adjunction requires the following equalities:
4050:
3559:
3466:
3446:
3426:
3074:
2908:
2373:
1507:
710:
344:
281:
257:
237:
4340:
5413:) entangled measurement between Y and another qubit X, then this qubit X will be teleported from Alice to Bob: quantum teleportation is an identity morphism.
4438:
2071:
4672:{\displaystyle d\otimes {\text{dom}}(d')\ \circ \ {\text{cod}}(d)\otimes d'\quad =\quad {\text{dom}}(d)\otimes d'\ \circ \ d\otimes {\text{cod}}(d')}
853:
3471:
2586:
6211:. Lecture Notes in Computer Science. Vol. CONCUR 2014 - Concurrency Theory - 25th International Conference. Rome, Italy. pp. 435–450.
6248:
4055:
2290:
4702:
diagrams, i.e. whenever the plane graphs have no more than one connected component which is not connected to the domain or codomain and the
3889:. In fact, the explicit list of layers is redundant, it is enough to specify the length of the type to the left of each layer, known as the
1137:
1075:
5976:
Fong, Brendan; Spivak, David I.; Tuyéras, Rémy (2019-05-01). "Backprop as
Functor: A compositional perspective on supervised learning".
3243:
the composition of two diagrams as their vertical concatenation with the codomain of the first identified with the domain of the second,
644:
is a string diagram representing the sequential composition of processes, it is drawn as the vertical concatenation of the two diagrams.
536:
is a string diagram representing the parallel composition of processes, it is drawn as the horizontal concatenation of the two diagrams,
6156:
Fritz, Tobias (August 2020). "A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics".
3668:
6384:
2457:
2138:
1943:
3925:
1315:
6669:
5276:{\displaystyle {\begin{aligned}(\varepsilon F)\circ F(\eta )&=1_{F}\\G(\varepsilon )\circ (\eta G)&=1_{G}\end{aligned}}}
5008:
2386:
4968:
3798:
1242:
392:
is a string diagram representing the process which does nothing to its input system, it is drawn as a bunch of parallel wires,
6224:
5758:
3570:
2675:
1589:. Intuitively, once the image of generating objects and arrows are given, the image of every diagram they generate is fixed.
5493:
5605:
5583:
5455:
5303:
3743:
1787:
4531:
Two diagrams are equal (up to the axioms of monoidal categories) whenever they are in the same equivalence class of the
1608:
1019:
963:
2533:
6016:
4286:
4238:
4783:
5118:
are respectively the unit and the counit. The string diagrams corresponding to these natural transformations are:
5086:
1689:
7031:
6356:
4692:
2135:
of the string diagram, i.e. the list of edges that are connected to the top and bottom boundary. The other nodes
3205:
1724:
4876:
6377:
5574:
5375:
3622:
1410:
83:
56:
480:
7036:
6581:
6536:
5569:
5051:
926:
352:
3028:
7010:
6950:
5596:
5483:
5473:
4850:
Duality between commutative diagrams (on the left hand side) and string diagrams (on the right hand side)
4703:
1903:
1843:
94:. While the diagrams in these first articles were hand-drawn, the advent of typesetting software such as
6659:
3401:{\displaystyle (x,f,y)\in \Sigma _{0}^{\star }\times \Sigma _{1}\times \Sigma _{0}^{\star }=:L(\Sigma )}
1512:
7000:
6786:
6650:
6558:
5635:
5534:
5507:
5447:
4684:
1551:
1199:
52:
2825:
2256:
2001:
776:
6959:
6603:
6541:
6464:
3120:
609:
4944:
4920:
4207:
3174:
3079:
1309:
to its underlying morphism of signatures, i.e. it forgets the identity, composition and tensor. The
6990:
6946:
6551:
6370:
5587:
4687:
for free monoidal categories, i.e. deciding whether two given diagrams are equal, can be solved in
3268:
3260:
2454:
is injective, i.e. no two inner nodes are at the same height. In that case, one can define a list
117:
are arguably the oldest form of string diagrams, they are interpreted in the monoidal category of
6546:
6528:
6318:
6158:
5997:
Ghani, Neil; Hedges, Jules; Winschel, Viktor; Zahn, Philipp (2018). "Compositional Game Theory".
5789:
5503:
2034:
1760:
1729:
1465:
1415:
815:
745:
430:
32:
6262:
Bonchi, Filippo; Seeber, Jens; Sobocinski, Pawel (2018-04-20). "Graphical
Conjunctive Queries".
4177:
3899:
664:
6753:
6519:
6499:
6422:
5046:
3272:
114:
110:
4736:
574:
542:
398:
294:
183:
6635:
6474:
5430:
5390:
3285:
2513:
1663:
1445:
1392:
721:
3866:
2955:
2858:
2749:
1877:
6447:
6442:
5738:
5544:
3536:
2990:
2913:
2784:
2218:
604:
150:
5727:, Lecture Notes in Physics, vol. 813, Berlin, Heidelberg: Springer, pp. 95–172,
3279:, which are in bijection with the labeled generic progressive plane graphs defined above.
8:
6791:
6739:
6665:
6469:
4532:
3532:
59:
where the axioms of quantum theory are expressed in the language of monoidal categories.
24:
5742:
4844:
4723:
4423:{\displaystyle d\otimes d'=d\otimes {\text{dom}}(d')\ \circ \ {\text{cod}}(d)\otimes d'}
6645:
6640:
6622:
6504:
6479:
6284:
6263:
6242:
6230:
6185:
6167:
6138:
6071:
6053:
6022:
5977:
5764:
5728:
5677:
Hotz, Günter (1965). "Eine
Algebraisierung des Syntheseproblems von Schaltkreisen I.".
5626:
5529:
4035:
3544:
3451:
3431:
3411:
3059:
2893:
2358:
1492:
695:
329:
266:
242:
222:
71:
5857:
6954:
6891:
6879:
6781:
6706:
6701:
6655:
6437:
6432:
6220:
6189:
6142:
6130:
6122:
6012:
5802:
5768:
5754:
5630:
5564:
5559:
5513:
5426:
4521:{\textstyle d\otimes d'={\text{dom}}(d)\otimes d'\ \circ \ d\otimes {\text{cod}}(d')}
2662:
from inner nodes to generating arrows, in a way compatible with domain and codomain.
1598:
1237:
475:
347:
158:
134:
126:
106:
6234:
6026:
6915:
6801:
6776:
6711:
6696:
6691:
6630:
6459:
6427:
6212:
6177:
6112:
6102:
6075:
6063:
6004:
5853:
5816:
5798:
5746:
5443:
5439:
5398:
4871:
4695:
2127:{\displaystyle {\text{dom}}(\Gamma ),{\text{cod}}(\Gamma )\in \Gamma _{1}^{\star }}
1386:
1306:
154:
6827:
6393:
6216:
6107:
6090:
5639:
5579:
5487:
5406:
4688:
3256:
1684:
1658:
166:
122:
79:
6204:
5911:"Normalization for planar string diagrams and a quadratic equivalence algorithm"
5750:
5506:
with 4-dimensional diagrams where the edges are undirected, a generalisation of
5496:
with 3-dimensional diagrams where the edges are undirected, a generalisation of
5363:
A monoidal category where every object has a left and right adjoint is called a
1548:, which by freeness is uniquely determined by a morphism of monoidal signatures
70:
gave the first mathematical definition of string diagrams in order to formalise
6864:
6859:
6843:
6796:
6716:
6325:
6091:"Reasoning about conscious experience with axiomatic and graphical mathematics"
5657:
5645:
5364:
2670:
48:
44:
6309:
6181:
6067:
6000:
Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in
Computer Science
5720:
87:
82:
also described as a precursor. They were later characterised as the arrows of
67:
7025:
6854:
6686:
6563:
6489:
6126:
5554:
5549:
5410:
5386:
4334:
the composition of two diagrams as the concatenation of their list of layers,
1681:
904:{\displaystyle {\text{dom}},{\text{cod}}:\Sigma _{1}\to \Sigma _{0}^{\star }}
162:
75:
36:
6008:
5525:
String diagrams have been used to formalise the following objects of study.
5294:
3524:{\displaystyle {\text{dom}},{\text{cod}}:L(\Sigma )\to \Sigma _{0}^{\star }}
6608:
6509:
6134:
5649:
5621:
5486:
with 4-dimensional diagrams where edges can cross, a generalisation of the
4824:
is represented by an intersection of strings (the strings corresponding to
2655:{\displaystyle v_{1}:\Gamma _{0}-\{a,b\}\times \mathbb {R} \to \Sigma _{1}}
1837:
1602:
1310:
28:
5393:
protocol. The unit and counit of the adjunction are an abstraction of the
5150:
6869:
6849:
6721:
6591:
5787:
Joyal, André; Street, Ross (1991). "The geometry of tensor calculus, I".
5591:
5539:
5497:
5477:
4167:{\displaystyle d\otimes z=(x_{1},f_{1},y_{1}z)\dots (x_{n},f_{n},y_{n}z)}
689:
659:
118:
91:
5138:
6901:
6839:
6452:
6117:
5842:"A categorical interpretation of CS Peirce's propositional logic Alpha"
5653:
5394:
2346:{\displaystyle {\text{dom}}(f),{\text{cod}}(f)\in \Gamma _{1}^{\star }}
40:
5126:
6895:
6586:
6089:
Signorelli, Camilo Miguel; Wang, Quanlong; Coecke, Bob (2021-10-01).
5617:
146:
20:
6203:
Bonchi, Filippo; Sobociński, Pawel; Zanasi, Fabio (September 2014).
6964:
6596:
6494:
6337:
6289:
6268:
6172:
5982:
5600:
5451:
1189:{\displaystyle {\text{cod}}\circ F_{1}\ =\ F_{0}\circ {\text{cod}}}
1127:{\displaystyle {\text{dom}}\circ F_{1}\ =\ F_{0}\circ {\text{dom}}}
142:
99:
74:. However, the invention of string diagrams is usually credited to
6362:
6058:
5733:
1305:
which sends a monoidal category to its underlying signature and a
6934:
6924:
6573:
6484:
3730:{\displaystyle {\text{layers}}(d)=d_{1}\dots d_{n}\in L(\Sigma )}
1072:
which is compatible with the domain and codomain, i.e. such that
213:
5817:"Categories: History of string diagrams (thread, 2017may02-...)"
287:, one may generate the set of all string diagrams by induction:
6929:
6351:
5620:, a generalisation of string diagrams used to denote proofs in
5402:
5389:
is rigid, this fact underlies the proof of correctness for the
3255:
While the geometric definition makes explicit the link between
2496:{\displaystyle {\text{boxes}}(\Gamma )\in \Gamma _{0}^{\star }}
2193:{\displaystyle f\in \Gamma _{0}\ -\ \{a,b\}\times \mathbb {R} }
1991:{\displaystyle x\in \Gamma \ \cap \ \mathbb {R} \times \{a,b\}}
260:
5877:
International
Conference on Theory and Application of Diagrams
4769:—separating the plane in two (the right part corresponding to
4025:{\displaystyle d=(x_{1},f_{1},y_{1})\dots (x_{n},f_{n},y_{n})}
6811:
4337:
the tensor of two diagrams as the composition of whiskerings
3246:
the tensor of two diagrams as their horizontal concatenation.
95:
5896:
Unpublished
Manuscript, Available from Ross Street's Website
5629:, a precursor of string diagrams used to denote formulae in
1378:{\displaystyle C_{-}:\mathbf {MonSig} \to \mathbf {MonCat} }
161:, invented independently from the one-dimensional syntax of
6329:
5721:"Physics, Topology, Logic and Computation: A Rosetta Stone"
4052:
is defined as the concatenation to the right of each layer
5038:{\displaystyle G:{\mathcal {C}}\rightarrow {\mathcal {D}}}
2447:{\displaystyle \Gamma _{0}-\{a,b\}\times \mathbb {R} \to }
2376:, given by the list of edges that have source and target.
283:. Starting from a collection of wires and boxes, called a
102:
made the publication of string diagrams more wide-spread.
5934:"A survey of graphical languages for monoidal categories"
5367:. String diagrams for rigid categories can be defined as
4998:{\displaystyle F:{\mathcal {C}}\leftarrow {\mathcal {D}}}
3856:{\displaystyle {\text{cod}}(d_{i})={\text{dom}}(d_{i+1})}
1757:, each homeomorphic to an open interval with boundary in
1298:{\displaystyle U:\mathbf {MonCat} \to \mathbf {MonSig} }
5996:
3609:{\displaystyle {\text{dom}}(d)\in \Sigma _{0}^{\star }}
3165:
equivalence classes of labeled progressive plane graphs
2735:{\displaystyle h:\Gamma \times \to \times \mathbb {R} }
133:
of Peirce's existential graphs can be axiomatised as a
6261:
6202:
5468:
Survey of graphical languages for monoidal categories.
4441:
3539:, with the types as vertices and the layers as edges.
1790:
779:
6340:, a Python toolkit for computing with string diagrams
5679:
5450:. This observation has led to the development of the
5433:
protocol as modeled in categorical quantum mechanics.
5306:
5172:
5089:
5054:
5011:
4971:
4947:
4923:
4879:
4786:
4739:
4545:
4343:
4289:
4241:
4210:
4180:
4058:
4038:
3928:
3902:
3869:
3801:
3746:
3671:
3625:
3573:
3547:
3474:
3454:
3434:
3414:
3312:
3288:
3208:
3177:
3123:
3082:
3062:
3031:
2993:
2958:
2916:
2896:
2861:
2828:
2787:
2752:
2678:
2589:
2536:
2516:
2460:
2389:
2361:
2293:
2259:
2221:
2141:
2074:
2037:
2004:
1946:
1906:
1880:
1846:
1763:
1732:
1692:
1666:
1611:
1554:
1515:
1495:
1468:
1448:
1418:
1395:
1389:
to the forgetful functor, sends a monoidal signature
1318:
1245:
1202:
1140:
1078:
1022:
966:
929:
856:
818:
748:
724:
698:
667:
612:
577:
545:
483:
433:
401:
355:
332:
297:
269:
245:
225:
186:
6088:
6042:"Picturing classical and quantum Bayesian inference"
1826:{\textstyle \Gamma -\Gamma _{0}=\coprod \Gamma _{1}}
259:
at the bottom, which represent the input and output
5461:
5350:{\displaystyle (\varepsilon F)\circ F(\eta )=1_{F}}
3788:{\displaystyle {\text{dom}}(d_{1})={\text{dom}}(d)}
3025:. Deformations induce an equivalence relation with
2952:A deformation is progressive (generic, labeled) if
5429:An illustration of the diagrammatic calculus: the
5349:
5275:
5110:
5075:
5037:
4997:
4957:
4933:
4909:
4816:
4757:
4671:
4520:
4422:
4323:
4275:
4227:
4192:
4166:
4044:
4024:
3914:
3881:
3855:
3787:
3729:
3654:
3608:
3553:
3523:
3460:
3440:
3420:
3400:
3294:
3232:
3194:
3155:
3109:
3068:
3048:
3017:
2979:
2940:
2902:
2882:
2847:
2811:
2773:
2734:
2654:
2575:
2522:
2495:
2446:
2367:
2345:
2278:
2245:
2192:
2126:
2050:
2031:and belongs to the closure of exactly one edge in
2023:
1990:
1932:
1892:
1866:
1825:
1776:
1745:
1711:
1672:
1649:
1581:
1540:
1501:
1481:
1454:
1431:
1401:
1377:
1297:
1225:
1188:
1126:
1064:
1008:
952:
903:
831:
797:
761:
730:
704:
680:
636:
595:
563:
528:
466:
419:
384:
338:
315:
275:
251:
231:
204:
5975:
5476:with 3-dimensional diagrams, a generalisation of
4714:The idea is to represent structures of dimension
1650:{\displaystyle (\Gamma ,\Gamma _{0},\Gamma _{1})}
1065:{\displaystyle F_{1}:\Sigma _{1}\to \Sigma '_{1}}
1009:{\displaystyle F_{0}:\Sigma _{0}\to \Sigma '_{0}}
19:are a formal graphical language for representing
7023:
6283:Riley, Mitchell (2018). "Categories of optics".
5870:
5516:where every diagram has a horizontal reflection.
5371:plane graphs, i.e. the edges can bend backward.
3468:on the right. Layers have a domain and codomain
3202:as a set of parallel edges labeled by some type
2576:{\displaystyle v_{0}:\Gamma _{1}\to \Sigma _{0}}
6205:"A Categorical Semantics of Signal Flow Graphs"
5438:The same equation appears in the definition of
4730:an object is represented by a portion of plane,
2503:of the inner nodes ordered from top to bottom.
692:, i.e. the set of lists with elements in a set
141:are unary operators on homsets that axiomatise
35:. When interpreted in the monoidal category of
6039:
5963:International Conference on Concurrency Theory
5908:
5873:"Compositional diagrammatic first-order logic"
5719:Baez, J.; Stay, M. (2011), Coecke, Bob (ed.),
5702:Combinatorial Mathematics and Its Applications
5698:"Applications of negative dimensional tensors"
5642:, two precursors of string diagrams in physics
4765:is represented by a vertical segment—called a
4324:{\displaystyle {\text{dom}}({\text{id}}(x))=x}
4276:{\displaystyle {\text{len}}({\text{id}}(x))=0}
2530:if it comes equipped with a pair of functions
6378:
4828:above the link, the strings corresponding to
4817:{\displaystyle \alpha :f\Rightarrow g:A\to B}
3267:is necessary to formalise string diagrams in
919:to each box, i.e. the input and output types.
5839:
5300:Diagrammatic representation of the equality
5111:{\displaystyle \varepsilon :FG\rightarrow I}
4709:
2628:
2616:
2415:
2403:
2179:
2167:
1985:
1973:
1712:{\displaystyle \Gamma _{0}\subseteq \Gamma }
1489:. The interpretation in a monoidal category
1233:of monoidal signatures and their morphisms.
5889:
5871:Haydon, Nathan; Sobociński, Pawe\l (2020).
5786:
5648:, the interpretation of string diagrams in
1462:) are arrows in the free monoidal category
7006:
6996:
6752:
6385:
6371:
6247:: CS1 maint: location missing publisher (
5840:Brady, Geraldine; Trimble, Todd H (2000).
2987:is progressive (generic, labeled) for all
1900:is a finite topological graph embedded in
6288:
6267:
6171:
6116:
6106:
6057:
5981:
5959:"Retracing some paths in process algebra"
5909:Vicary, Jamie; Delpeuch, Antonin (2022).
5732:
5401:respectively. If Alice and Bob share two
3531:defined in the obvious way. This forms a
3233:{\displaystyle x\in \Sigma _{0}^{\star }}
2728:
2635:
2422:
2186:
1966:
1908:
1860:
6040:Coecke, Bob; Spekkens, Robert W (2012).
5956:
5931:
5718:
4910:{\displaystyle (F,G,\eta ,\varepsilon )}
5695:
5520:
3655:{\displaystyle {\text{len}}(d)=n\geq 0}
3561:is encoded as a path in this multigraph
7024:
1442:String diagrams (with generators from
529:{\displaystyle f\otimes f':xx'\to yy'}
6751:
6404:
6366:
6282:
6155:
5076:{\displaystyle \eta :I\rightarrow GF}
4865:
4174:and symmetrically for the whiskering
2583:from edges to generating objects and
953:{\displaystyle F:\Sigma \to \Sigma '}
773:, the lists of generating objects in
385:{\displaystyle {\text{id}}(x):x\to x}
55:. This has led to the development of
5892:"Planar diagrams and tensor algebra"
5676:
3049:{\displaystyle \Gamma \sim \Gamma '}
6392:
6344:
5915:Logical Methods in Computer Science
5890:Joyal, André; Street, Ross (1988).
5846:Journal of Pure and Applied Algebra
5606:Quantum natural language processing
5584:measurement-based quantum computing
5456:quantum natural language processing
1933:{\displaystyle \mathbb {R} \times }
1867:{\displaystyle a,b\in \mathbb {R} }
1509:is a defined by a monoidal functor
13:
5925:
5030:
5020:
4990:
4980:
4950:
4926:
4200:on the left. One can then define:
3721:
3592:
3507:
3497:
3392:
3369:
3356:
3338:
3289:
3216:
3146:
3104:
3039:
3032:
2836:
2685:
2643:
2604:
2564:
2551:
2517:
2479:
2469:
2391:
2329:
2267:
2149:
2110:
2100:
2083:
2039:
2012:
1953:
1814:
1798:
1791:
1765:
1734:
1706:
1694:
1667:
1635:
1622:
1615:
1561:
1541:{\displaystyle F:C_{\Sigma }\to D}
1527:
1474:
1449:
1424:
1396:
1050:
1037:
994:
981:
943:
936:
887:
874:
820:
781:
750:
725:
14:
7048:
6301:
1582:{\displaystyle F:\Sigma \to U(D)}
1226:{\displaystyle \mathbf {MonSig} }
923:A morphism of monoidal signature
798:{\textstyle \Sigma _{0}^{\star }}
539:for each pair of string diagrams
395:for each pair of string diagrams
7005:
6995:
6986:
6985:
6738:
6405:
6350:
6321:from the original on 2021-12-19.
6209:CONCUR 2014 – Concurrency Theory
5462:Hierarchy of graphical languages
5442:where it captures the notion of
5418:
5293:
5149:
5137:
5125:
4843:
3250:
2848:{\displaystyle x\in \Gamma _{0}}
2279:{\displaystyle e\in \Gamma _{1}}
2024:{\displaystyle x\in \Gamma _{0}}
1601:, also called a one-dimensional
1371:
1368:
1365:
1362:
1359:
1356:
1348:
1345:
1342:
1339:
1336:
1333:
1291:
1288:
1285:
1282:
1279:
1276:
1268:
1265:
1262:
1259:
1256:
1253:
1219:
1216:
1213:
1210:
1207:
1204:
6276:
6255:
6196:
6149:
6082:
6033:
5990:
5969:
5950:
4609:
4605:
3156:{\displaystyle h(-,1)=\Gamma '}
2215:, when the vertical projection
637:{\displaystyle g\circ f:x\to z}
145:. This makes string diagrams a
31:. They are a prominent tool in
27:, or more generally 2-cells in
5902:
5883:
5864:
5833:
5809:
5780:
5712:
5689:
5670:
5331:
5325:
5316:
5307:
5249:
5240:
5234:
5228:
5201:
5195:
5186:
5177:
5156:String diagram of the identity
5102:
5064:
5025:
4985:
4958:{\displaystyle {\mathcal {D}}}
4934:{\displaystyle {\mathcal {C}}}
4904:
4880:
4808:
4796:
4749:
4666:
4655:
4621:
4615:
4591:
4585:
4568:
4557:
4515:
4504:
4470:
4464:
4406:
4400:
4383:
4372:
4312:
4309:
4303:
4295:
4264:
4261:
4255:
4247:
4228:{\displaystyle {\text{id}}(x)}
4222:
4216:
4161:
4119:
4113:
4071:
4019:
3980:
3974:
3935:
3850:
3831:
3820:
3807:
3782:
3776:
3765:
3752:
3724:
3718:
3683:
3677:
3637:
3631:
3585:
3579:
3503:
3500:
3494:
3395:
3389:
3331:
3313:
3195:{\displaystyle {\text{id}}(x)}
3189:
3183:
3139:
3127:
3110:{\displaystyle h(-,0)=\Gamma }
3098:
3086:
3012:
3000:
2974:
2962:
2935:
2923:
2877:
2865:
2806:
2794:
2781:defines a plane graph for all
2768:
2756:
2721:
2709:
2706:
2703:
2691:
2639:
2560:
2472:
2466:
2441:
2429:
2426:
2322:
2316:
2305:
2299:
2240:
2228:
2225:
2103:
2097:
2086:
2080:
1927:
1915:
1644:
1612:
1576:
1570:
1564:
1532:
1352:
1272:
1046:
990:
939:
883:
628:
587:
555:
512:
453:
411:
376:
367:
361:
307:
196:
1:
5858:10.1016/S0022-4049(98)00179-0
5663:
5575:Categorical quantum mechanics
5570:Bidirectional transformations
5484:Symmetric monoidal categories
5376:categorical quantum mechanics
5286:The first one is depicted as
3056:if and only if there is some
2506:A progressive plane graph is
2383:when the vertical projection
648:
57:categorical quantum mechanics
47:, string diagrams are called
6217:10.1007/978-3-662-44584-6_30
6108:10.1016/j.concog.2021.103168
5940:, Springer, pp. 289–355
5803:10.1016/0001-8708(91)90003-P
5144:String diagram of the counit
2253:is injective for every edge
1592:
653:
177:String diagrams are made of
172:
7:
6680:Constructions on categories
6095:Consciousness and Cognition
5751:10.1007/978-3-642-12821-9_2
5611:
5597:Natural language processing
5474:Braided monoidal categories
4860:
4718:by structures of dimension
2051:{\displaystyle \Gamma _{1}}
1777:{\displaystyle \Gamma _{0}}
1746:{\displaystyle \Gamma _{1}}
1482:{\displaystyle C_{\Sigma }}
1432:{\displaystyle C_{\Sigma }}
1196:. Thus we get the category
832:{\displaystyle \Sigma _{1}}
762:{\displaystyle \Sigma _{0}}
467:{\displaystyle f':x'\to y'}
263:being processed by the box
10:
7053:
6787:Higher-dimensional algebra
5938:New structures for physics
5725:New Structures for Physics
5636:Penrose graphical notation
5535:Artificial neural networks
5508:Penrose graphical notation
5448:natural language semantics
5132:String diagram of the unit
4193:{\displaystyle z\otimes d}
3915:{\displaystyle d\otimes z}
3167:. Indeed, one can define:
2890:is an inner node for some
681:{\displaystyle X^{\star }}
62:
53:Penrose graphical notation
6981:
6914:
6878:
6826:
6819:
6770:
6760:
6747:
6736:
6679:
6621:
6572:
6527:
6518:
6415:
6411:
6400:
6182:10.1016/j.aim.2020.107239
6068:10.1007/s11229-011-9917-5
5957:Abramsky, Samson (1996).
5504:Compact closed categories
4710:Extension to 2-categories
3448:in the middle and a type
3282:Fix a monoidal signature
2058:. Such points are called
1840:between two real numbers
239:coming in at the top and
5932:Selinger, Peter (2010),
5588:quantum error correction
4758:{\displaystyle f:A\to B}
4691:. The interchanger is a
3269:computer algebra systems
3265:combinatorial definition
3261:low-dimensional topology
2510:by a monoidal signature
596:{\displaystyle g:y\to z}
564:{\displaystyle f:x\to y}
420:{\displaystyle f:x\to y}
316:{\displaystyle f:x\to y}
205:{\displaystyle f:x\to y}
86:in a seminal article by
84:free monoidal categories
6597:Cokernels and quotients
6520:Universal constructions
6159:Advances in Mathematics
6009:10.1145/3209108.3209165
5790:Advances in Mathematics
5696:Penrose, Roger (1971).
5378:, this is known as the
5047:natural transformations
4917:between two categories
4704:Eckmann–Hilton argument
3563:, i.e. it is given by:
3306:is defined as a triple
3295:{\displaystyle \Sigma }
3271:and use them to define
2523:{\displaystyle \Sigma }
1673:{\displaystyle \Gamma }
1455:{\displaystyle \Sigma }
1402:{\displaystyle \Sigma }
960:is a pair of functions
731:{\displaystyle \Sigma }
326:for each list of wires
33:applied category theory
7032:Higher category theory
6754:Higher category theory
6500:Natural transformation
5409:and Alice performs a (
5351:
5277:
5112:
5077:
5039:
4999:
4959:
4935:
4911:
4818:
4759:
4673:
4522:
4424:
4325:
4277:
4229:
4194:
4168:
4046:
4026:
3916:
3883:
3882:{\displaystyle i<n}
3857:
3789:
3731:
3656:
3610:
3555:
3525:
3462:
3442:
3422:
3402:
3296:
3273:computational problems
3234:
3196:
3163:. String diagrams are
3157:
3111:
3070:
3050:
3019:
2981:
2980:{\displaystyle h(-,t)}
2942:
2904:
2884:
2883:{\displaystyle h(x,t)}
2849:
2813:
2775:
2774:{\displaystyle h(-,t)}
2736:
2656:
2577:
2524:
2497:
2448:
2369:
2347:
2280:
2247:
2194:
2128:
2052:
2025:
1992:
1940:such that every point
1934:
1894:
1893:{\displaystyle a<b}
1868:
1827:
1778:
1747:
1713:
1674:
1651:
1583:
1542:
1503:
1483:
1456:
1433:
1411:free monoidal category
1403:
1379:
1299:
1227:
1190:
1128:
1066:
1010:
954:
905:
833:
799:
763:
732:
706:
682:
638:
597:
565:
530:
468:
421:
386:
340:
317:
277:
253:
233:
206:
115:Charles Sanders Peirce
111:diagrammatic reasoning
5431:quantum teleportation
5391:quantum teleportation
5352:
5278:
5113:
5078:
5040:
5000:
4960:
4936:
4912:
4819:
4760:
4674:
4523:
4425:
4326:
4278:
4230:
4204:the identity diagram
4195:
4169:
4047:
4027:
3917:
3884:
3858:
3790:
3732:
3657:
3611:
3556:
3526:
3463:
3443:
3423:
3403:
3297:
3235:
3197:
3171:the identity diagram
3158:
3112:
3071:
3051:
3020:
3018:{\displaystyle t\in }
2982:
2943:
2941:{\displaystyle t\in }
2905:
2885:
2850:
2814:
2812:{\displaystyle t\in }
2776:
2737:
2669:of plane graphs is a
2657:
2578:
2525:
2498:
2449:
2370:
2348:
2281:
2248:
2246:{\displaystyle e\to }
2195:
2129:
2053:
2026:
1993:
1935:
1895:
1869:
1828:
1779:
1748:
1714:
1675:
1652:
1584:
1543:
1504:
1484:
1457:
1434:
1404:
1380:
1300:
1228:
1191:
1129:
1067:
1011:
955:
906:
834:
800:
764:
733:
707:
683:
639:
598:
566:
531:
469:
422:
387:
341:
318:
278:
254:
234:
207:
6623:Algebraic categories
6359:at Wikimedia Commons
6308:TheCatsters (2007).
6003:. pp. 472–481.
5879:. Springer: 402–418.
5545:Bayesian probability
5521:List of applications
5304:
5170:
5087:
5052:
5009:
4969:
4945:
4921:
4877:
4784:
4773:and the left one to
4737:
4543:
4439:
4341:
4287:
4239:
4208:
4178:
4056:
4036:
3926:
3900:
3867:
3799:
3744:
3669:
3623:
3571:
3545:
3472:
3452:
3432:
3412:
3310:
3286:
3206:
3175:
3121:
3080:
3060:
3029:
2991:
2956:
2914:
2910:it is inner for all
2894:
2859:
2826:
2785:
2750:
2676:
2587:
2534:
2514:
2458:
2387:
2359:
2291:
2257:
2219:
2139:
2072:
2035:
2002:
1944:
1904:
1878:
1844:
1788:
1761:
1730:
1725:connected components
1690:
1664:
1609:
1552:
1513:
1493:
1466:
1446:
1416:
1393:
1316:
1243:
1200:
1138:
1076:
1020:
964:
927:
854:
850:a pair of functions
816:
777:
746:
722:
696:
665:
610:
575:
543:
481:
431:
399:
353:
330:
323:is a string diagram,
295:
267:
243:
223:
184:
7037:Monoidal categories
6792:Homotopy hypothesis
6470:Commutative diagram
5743:2011LNP...813...95B
5565:Conjunctive queries
5005:is left adjoint of
4533:congruence relation
3605:
3533:directed multigraph
3520:
3428:on the left, a box
3382:
3351:
3229:
2492:
2354:of each inner node
2342:
2123:
1061:
1005:
900:
794:
72:electronic circuits
25:monoidal categories
6505:Universal property
5627:Existential graphs
5560:Signal-flow graphs
5530:Concurrency theory
5374:In the context of
5347:
5273:
5271:
5108:
5073:
5035:
4995:
4955:
4931:
4907:
4866:The snake equation
4814:
4755:
4700:boundary connected
4669:
4518:
4420:
4321:
4273:
4225:
4190:
4164:
4042:
4022:
3912:
3879:
3853:
3785:
3727:
3652:
3606:
3591:
3551:
3535:, also known as a
3521:
3506:
3458:
3438:
3418:
3398:
3368:
3337:
3292:
3230:
3215:
3192:
3153:
3107:
3066:
3046:
3015:
2977:
2938:
2900:
2880:
2845:
2809:
2771:
2732:
2652:
2573:
2520:
2493:
2478:
2444:
2365:
2343:
2328:
2276:
2243:
2190:
2124:
2109:
2062:, they define the
2048:
2021:
1988:
1930:
1890:
1864:
1823:
1774:
1743:
1709:
1670:
1647:
1579:
1538:
1499:
1479:
1452:
1429:
1399:
1375:
1295:
1223:
1186:
1124:
1062:
1049:
1006:
993:
950:
901:
886:
829:
795:
780:
771:generating objects
759:
728:
717:monoidal signature
702:
678:
634:
593:
561:
526:
464:
417:
382:
336:
313:
273:
249:
229:
212:, which represent
202:
107:existential graphs
7019:
7018:
6977:
6976:
6973:
6972:
6955:monoidal category
6910:
6909:
6782:Enriched category
6734:
6733:
6730:
6729:
6707:Quotient category
6702:Opposite category
6617:
6616:
6355:Media related to
6311:String diagrams 1
6226:978-3-662-44583-9
5965:. Springer: 1–17.
5760:978-3-642-12821-9
5631:first-order logic
5514:Dagger categories
5494:Ribbon categories
5440:pregroup grammars
4698:on the subset of
4653:
4643:
4637:
4613:
4583:
4579:
4573:
4555:
4535:generated by the
4502:
4492:
4486:
4462:
4398:
4394:
4388:
4370:
4301:
4293:
4253:
4245:
4214:
4045:{\displaystyle z}
3829:
3805:
3774:
3750:
3675:
3629:
3616:as starting point
3577:
3554:{\displaystyle d}
3541:A string diagram
3486:
3478:
3461:{\displaystyle y}
3441:{\displaystyle f}
3421:{\displaystyle x}
3181:
3069:{\displaystyle h}
2903:{\displaystyle t}
2464:
2379:A plane graph is
2368:{\displaystyle f}
2314:
2297:
2207:A plane graph is
2166:
2160:
2095:
2078:
1964:
1958:
1599:topological graph
1502:{\displaystyle D}
1238:forgetful functor
1184:
1167:
1161:
1144:
1122:
1105:
1099:
1082:
868:
860:
841:generating arrows
705:{\displaystyle X}
359:
339:{\displaystyle x}
276:{\displaystyle f}
252:{\displaystyle y}
232:{\displaystyle x}
216:, with a list of
159:first-order logic
135:Frobenius algebra
131:lines of identity
127:Cartesian product
7044:
7009:
7008:
6999:
6998:
6989:
6988:
6824:
6823:
6802:Simplex category
6777:Categorification
6768:
6767:
6749:
6748:
6742:
6712:Product category
6697:Kleisli category
6692:Functor category
6537:Terminal objects
6525:
6524:
6460:Adjoint functors
6413:
6412:
6402:
6401:
6387:
6380:
6373:
6364:
6363:
6354:
6322:
6316:
6315:(streamed video)
6295:
6294:
6292:
6280:
6274:
6273:
6271:
6259:
6253:
6252:
6246:
6238:
6200:
6194:
6193:
6175:
6153:
6147:
6146:
6120:
6110:
6086:
6080:
6079:
6061:
6037:
6031:
6030:
5994:
5988:
5987:
5985:
5973:
5967:
5966:
5954:
5948:
5947:
5946:
5945:
5929:
5923:
5922:
5906:
5900:
5899:
5887:
5881:
5880:
5868:
5862:
5861:
5837:
5831:
5830:
5828:
5827:
5813:
5807:
5806:
5784:
5778:
5777:
5776:
5775:
5736:
5716:
5710:
5709:
5693:
5687:
5686:
5674:
5640:Feynman diagrams
5580:Quantum circuits
5444:information flow
5422:
5399:Bell measurement
5385:The category of
5356:
5354:
5353:
5348:
5346:
5345:
5297:
5282:
5280:
5279:
5274:
5272:
5268:
5267:
5220:
5219:
5153:
5141:
5129:
5117:
5115:
5114:
5109:
5082:
5080:
5079:
5074:
5044:
5042:
5041:
5036:
5034:
5033:
5024:
5023:
5004:
5002:
5001:
4996:
4994:
4993:
4984:
4983:
4964:
4962:
4961:
4956:
4954:
4953:
4940:
4938:
4937:
4932:
4930:
4929:
4916:
4914:
4913:
4908:
4847:
4832:below the link).
4823:
4821:
4820:
4815:
4764:
4762:
4761:
4756:
4724:Poincaré duality
4706:does not apply.
4696:rewriting system
4678:
4676:
4675:
4670:
4665:
4654:
4651:
4641:
4635:
4634:
4614:
4611:
4604:
4584:
4581:
4577:
4571:
4567:
4556:
4553:
4527:
4525:
4524:
4519:
4514:
4503:
4500:
4490:
4484:
4483:
4463:
4460:
4455:
4429:
4427:
4426:
4421:
4419:
4399:
4396:
4392:
4386:
4382:
4371:
4368:
4357:
4330:
4328:
4327:
4322:
4302:
4299:
4294:
4291:
4282:
4280:
4279:
4274:
4254:
4251:
4246:
4243:
4234:
4232:
4231:
4226:
4215:
4212:
4199:
4197:
4196:
4191:
4173:
4171:
4170:
4165:
4157:
4156:
4144:
4143:
4131:
4130:
4109:
4108:
4096:
4095:
4083:
4082:
4051:
4049:
4048:
4043:
4031:
4029:
4028:
4023:
4018:
4017:
4005:
4004:
3992:
3991:
3973:
3972:
3960:
3959:
3947:
3946:
3921:
3919:
3918:
3913:
3888:
3886:
3885:
3880:
3862:
3860:
3859:
3854:
3849:
3848:
3830:
3827:
3819:
3818:
3806:
3803:
3794:
3792:
3791:
3786:
3775:
3772:
3764:
3763:
3751:
3748:
3736:
3734:
3733:
3728:
3711:
3710:
3698:
3697:
3676:
3673:
3661:
3659:
3658:
3653:
3630:
3627:
3615:
3613:
3612:
3607:
3604:
3599:
3578:
3575:
3560:
3558:
3557:
3552:
3530:
3528:
3527:
3522:
3519:
3514:
3487:
3484:
3479:
3476:
3467:
3465:
3464:
3459:
3447:
3445:
3444:
3439:
3427:
3425:
3424:
3419:
3407:
3405:
3404:
3399:
3381:
3376:
3364:
3363:
3350:
3345:
3301:
3299:
3298:
3293:
3239:
3237:
3236:
3231:
3228:
3223:
3201:
3199:
3198:
3193:
3182:
3179:
3162:
3160:
3159:
3154:
3152:
3116:
3114:
3113:
3108:
3075:
3073:
3072:
3067:
3055:
3053:
3052:
3047:
3045:
3024:
3022:
3021:
3016:
2986:
2984:
2983:
2978:
2947:
2945:
2944:
2939:
2909:
2907:
2906:
2901:
2889:
2887:
2886:
2881:
2854:
2852:
2851:
2846:
2844:
2843:
2818:
2816:
2815:
2810:
2780:
2778:
2777:
2772:
2741:
2739:
2738:
2733:
2731:
2661:
2659:
2658:
2653:
2651:
2650:
2638:
2612:
2611:
2599:
2598:
2582:
2580:
2579:
2574:
2572:
2571:
2559:
2558:
2546:
2545:
2529:
2527:
2526:
2521:
2502:
2500:
2499:
2494:
2491:
2486:
2465:
2462:
2453:
2451:
2450:
2445:
2425:
2399:
2398:
2374:
2372:
2371:
2366:
2352:
2350:
2349:
2344:
2341:
2336:
2315:
2312:
2298:
2295:
2285:
2283:
2282:
2277:
2275:
2274:
2252:
2250:
2249:
2244:
2199:
2197:
2196:
2191:
2189:
2164:
2158:
2157:
2156:
2133:
2131:
2130:
2125:
2122:
2117:
2096:
2093:
2079:
2076:
2057:
2055:
2054:
2049:
2047:
2046:
2030:
2028:
2027:
2022:
2020:
2019:
1997:
1995:
1994:
1989:
1969:
1962:
1956:
1939:
1937:
1936:
1931:
1911:
1899:
1897:
1896:
1891:
1873:
1871:
1870:
1865:
1863:
1832:
1830:
1829:
1824:
1822:
1821:
1806:
1805:
1783:
1781:
1780:
1775:
1773:
1772:
1752:
1750:
1749:
1744:
1742:
1741:
1718:
1716:
1715:
1710:
1702:
1701:
1679:
1677:
1676:
1671:
1656:
1654:
1653:
1648:
1643:
1642:
1630:
1629:
1588:
1586:
1585:
1580:
1547:
1545:
1544:
1539:
1531:
1530:
1508:
1506:
1505:
1500:
1488:
1486:
1485:
1480:
1478:
1477:
1461:
1459:
1458:
1453:
1438:
1436:
1435:
1430:
1428:
1427:
1408:
1406:
1405:
1400:
1384:
1382:
1381:
1376:
1374:
1351:
1328:
1327:
1307:monoidal functor
1304:
1302:
1301:
1296:
1294:
1271:
1232:
1230:
1229:
1224:
1222:
1195:
1193:
1192:
1187:
1185:
1182:
1177:
1176:
1165:
1159:
1158:
1157:
1145:
1142:
1133:
1131:
1130:
1125:
1123:
1120:
1115:
1114:
1103:
1097:
1096:
1095:
1083:
1080:
1071:
1069:
1068:
1063:
1057:
1045:
1044:
1032:
1031:
1015:
1013:
1012:
1007:
1001:
989:
988:
976:
975:
959:
957:
956:
951:
949:
910:
908:
907:
902:
899:
894:
882:
881:
869:
866:
861:
858:
838:
836:
835:
830:
828:
827:
805:are also called
804:
802:
801:
796:
793:
788:
768:
766:
765:
760:
758:
757:
737:
735:
734:
729:
711:
709:
708:
703:
687:
685:
684:
679:
677:
676:
643:
641:
640:
635:
602:
600:
599:
594:
570:
568:
567:
562:
535:
533:
532:
527:
525:
511:
497:
473:
471:
470:
465:
463:
452:
441:
426:
424:
423:
418:
391:
389:
388:
383:
360:
357:
345:
343:
342:
337:
322:
320:
319:
314:
282:
280:
279:
274:
258:
256:
255:
250:
238:
236:
235:
230:
211:
209:
208:
203:
155:deduction system
153:two-dimensional
143:logical negation
80:Feynman diagrams
7052:
7051:
7047:
7046:
7045:
7043:
7042:
7041:
7022:
7021:
7020:
7015:
6969:
6939:
6906:
6883:
6874:
6831:
6815:
6766:
6756:
6743:
6726:
6675:
6613:
6582:Initial objects
6568:
6514:
6407:
6396:
6394:Category theory
6391:
6347:
6326:String diagrams
6314:
6307:
6304:
6299:
6298:
6281:
6277:
6260:
6256:
6240:
6239:
6227:
6201:
6197:
6154:
6150:
6087:
6083:
6038:
6034:
6019:
5995:
5991:
5974:
5970:
5955:
5951:
5943:
5941:
5930:
5926:
5907:
5903:
5888:
5884:
5869:
5865:
5838:
5834:
5825:
5823:
5815:
5814:
5810:
5785:
5781:
5773:
5771:
5761:
5717:
5713:
5694:
5690:
5675:
5671:
5666:
5646:Tensor networks
5614:
5523:
5488:symmetric group
5464:
5436:
5435:
5434:
5428:
5423:
5407:entangled state
5369:non-progressive
5361:
5360:
5359:
5358:
5357:
5341:
5337:
5305:
5302:
5301:
5298:
5270:
5269:
5263:
5259:
5252:
5222:
5221:
5215:
5211:
5204:
5173:
5171:
5168:
5167:
5161:
5160:
5159:
5158:
5157:
5154:
5146:
5145:
5142:
5134:
5133:
5130:
5088:
5085:
5084:
5053:
5050:
5049:
5029:
5028:
5019:
5018:
5010:
5007:
5006:
4989:
4988:
4979:
4978:
4970:
4967:
4966:
4949:
4948:
4946:
4943:
4942:
4925:
4924:
4922:
4919:
4918:
4878:
4875:
4874:
4868:
4863:
4855:
4854:
4853:
4852:
4851:
4848:
4785:
4782:
4781:
4738:
4735:
4734:
4712:
4689:polynomial time
4658:
4650:
4627:
4610:
4597:
4580:
4560:
4552:
4544:
4541:
4540:
4507:
4499:
4476:
4459:
4448:
4440:
4437:
4436:
4412:
4395:
4375:
4367:
4350:
4342:
4339:
4338:
4298:
4290:
4288:
4285:
4284:
4250:
4242:
4240:
4237:
4236:
4211:
4209:
4206:
4205:
4179:
4176:
4175:
4152:
4148:
4139:
4135:
4126:
4122:
4104:
4100:
4091:
4087:
4078:
4074:
4057:
4054:
4053:
4037:
4034:
4033:
4013:
4009:
4000:
3996:
3987:
3983:
3968:
3964:
3955:
3951:
3942:
3938:
3927:
3924:
3923:
3901:
3898:
3897:
3868:
3865:
3864:
3838:
3834:
3826:
3814:
3810:
3802:
3800:
3797:
3796:
3771:
3759:
3755:
3747:
3745:
3742:
3741:
3706:
3702:
3693:
3689:
3672:
3670:
3667:
3666:
3626:
3624:
3621:
3620:
3600:
3595:
3574:
3572:
3569:
3568:
3546:
3543:
3542:
3515:
3510:
3483:
3475:
3473:
3470:
3469:
3453:
3450:
3449:
3433:
3430:
3429:
3413:
3410:
3409:
3377:
3372:
3359:
3355:
3346:
3341:
3311:
3308:
3307:
3287:
3284:
3283:
3257:category theory
3253:
3224:
3219:
3207:
3204:
3203:
3178:
3176:
3173:
3172:
3145:
3122:
3119:
3118:
3081:
3078:
3077:
3061:
3058:
3057:
3038:
3030:
3027:
3026:
2992:
2989:
2988:
2957:
2954:
2953:
2915:
2912:
2911:
2895:
2892:
2891:
2860:
2857:
2856:
2839:
2835:
2827:
2824:
2823:
2786:
2783:
2782:
2751:
2748:
2747:
2727:
2677:
2674:
2673:
2646:
2642:
2634:
2607:
2603:
2594:
2590:
2588:
2585:
2584:
2567:
2563:
2554:
2550:
2541:
2537:
2535:
2532:
2531:
2515:
2512:
2511:
2487:
2482:
2461:
2459:
2456:
2455:
2421:
2394:
2390:
2388:
2385:
2384:
2360:
2357:
2356:
2337:
2332:
2311:
2294:
2292:
2289:
2288:
2270:
2266:
2258:
2255:
2254:
2220:
2217:
2216:
2185:
2152:
2148:
2140:
2137:
2136:
2118:
2113:
2092:
2075:
2073:
2070:
2069:
2042:
2038:
2036:
2033:
2032:
2015:
2011:
2003:
2000:
1999:
1998:is also a node
1965:
1945:
1942:
1941:
1907:
1905:
1902:
1901:
1879:
1876:
1875:
1859:
1845:
1842:
1841:
1817:
1813:
1801:
1797:
1789:
1786:
1785:
1768:
1764:
1762:
1759:
1758:
1737:
1733:
1731:
1728:
1727:
1697:
1693:
1691:
1688:
1687:
1685:discrete subset
1665:
1662:
1661:
1659:Hausdorff space
1638:
1634:
1625:
1621:
1610:
1607:
1606:
1595:
1553:
1550:
1549:
1526:
1522:
1514:
1511:
1510:
1494:
1491:
1490:
1473:
1469:
1467:
1464:
1463:
1447:
1444:
1443:
1423:
1419:
1417:
1414:
1413:
1394:
1391:
1390:
1355:
1332:
1323:
1319:
1317:
1314:
1313:
1275:
1252:
1244:
1241:
1240:
1203:
1201:
1198:
1197:
1181:
1172:
1168:
1153:
1149:
1141:
1139:
1136:
1135:
1119:
1110:
1106:
1091:
1087:
1079:
1077:
1074:
1073:
1053:
1040:
1036:
1027:
1023:
1021:
1018:
1017:
997:
984:
980:
971:
967:
965:
962:
961:
942:
928:
925:
924:
911:which assign a
895:
890:
877:
873:
865:
857:
855:
852:
851:
823:
819:
817:
814:
813:
789:
784:
778:
775:
774:
753:
749:
747:
744:
743:
723:
720:
719:
697:
694:
693:
672:
668:
666:
663:
662:
656:
651:
611:
608:
607:
576:
573:
572:
544:
541:
540:
518:
504:
490:
482:
479:
478:
456:
445:
434:
432:
429:
428:
400:
397:
396:
356:
354:
351:
350:
331:
328:
327:
296:
293:
292:
268:
265:
264:
244:
241:
240:
224:
221:
220:
185:
182:
181:
175:
167:Begriffsschrift
65:
49:tensor networks
17:String diagrams
12:
11:
5:
7050:
7040:
7039:
7034:
7017:
7016:
7014:
7013:
7003:
6993:
6982:
6979:
6978:
6975:
6974:
6971:
6970:
6968:
6967:
6962:
6957:
6943:
6937:
6932:
6927:
6921:
6919:
6912:
6911:
6908:
6907:
6905:
6904:
6899:
6888:
6886:
6881:
6876:
6875:
6873:
6872:
6867:
6862:
6857:
6852:
6847:
6836:
6834:
6829:
6821:
6817:
6816:
6814:
6809:
6807:String diagram
6804:
6799:
6797:Model category
6794:
6789:
6784:
6779:
6774:
6772:
6765:
6764:
6761:
6758:
6757:
6745:
6744:
6737:
6735:
6732:
6731:
6728:
6727:
6725:
6724:
6719:
6717:Comma category
6714:
6709:
6704:
6699:
6694:
6689:
6683:
6681:
6677:
6676:
6674:
6673:
6663:
6653:
6651:Abelian groups
6648:
6643:
6638:
6633:
6627:
6625:
6619:
6618:
6615:
6614:
6612:
6611:
6606:
6601:
6600:
6599:
6589:
6584:
6578:
6576:
6570:
6569:
6567:
6566:
6561:
6556:
6555:
6554:
6544:
6539:
6533:
6531:
6522:
6516:
6515:
6513:
6512:
6507:
6502:
6497:
6492:
6487:
6482:
6477:
6472:
6467:
6462:
6457:
6456:
6455:
6450:
6445:
6440:
6435:
6430:
6419:
6417:
6409:
6408:
6398:
6397:
6390:
6389:
6382:
6375:
6367:
6361:
6360:
6357:String diagram
6346:
6345:External links
6343:
6342:
6341:
6335:
6323:
6303:
6302:External links
6300:
6297:
6296:
6275:
6254:
6225:
6195:
6148:
6081:
6052:(3): 651–696.
6032:
6017:
5989:
5968:
5949:
5924:
5901:
5882:
5863:
5852:(3): 213–239.
5832:
5808:
5779:
5759:
5711:
5688:
5668:
5667:
5665:
5662:
5661:
5660:
5658:tensor product
5643:
5633:
5624:
5613:
5610:
5609:
5608:
5603:
5594:
5577:
5572:
5567:
5562:
5557:
5555:Markov kernels
5552:
5547:
5542:
5537:
5532:
5522:
5519:
5518:
5517:
5511:
5501:
5491:
5481:
5463:
5460:
5454:framework and
5425:
5424:
5417:
5416:
5415:
5405:Y and Z in an
5387:Hilbert spaces
5380:snake equation
5365:rigid category
5344:
5340:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5299:
5292:
5291:
5290:
5289:
5288:
5284:
5283:
5266:
5262:
5258:
5255:
5253:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5223:
5218:
5214:
5210:
5207:
5205:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5175:
5155:
5148:
5147:
5143:
5136:
5135:
5131:
5124:
5123:
5122:
5121:
5120:
5107:
5104:
5101:
5098:
5095:
5092:
5072:
5069:
5066:
5063:
5060:
5057:
5032:
5027:
5022:
5017:
5014:
4992:
4987:
4982:
4977:
4974:
4952:
4928:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4867:
4864:
4862:
4859:
4849:
4842:
4841:
4840:
4839:
4838:
4834:
4833:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4778:
4754:
4751:
4748:
4745:
4742:
4731:
4711:
4708:
4668:
4664:
4661:
4657:
4649:
4646:
4640:
4633:
4630:
4626:
4623:
4620:
4617:
4608:
4603:
4600:
4596:
4593:
4590:
4587:
4576:
4570:
4566:
4563:
4559:
4551:
4548:
4517:
4513:
4510:
4506:
4498:
4495:
4489:
4482:
4479:
4475:
4472:
4469:
4466:
4458:
4454:
4451:
4447:
4444:
4432:
4431:
4418:
4415:
4411:
4408:
4405:
4402:
4391:
4385:
4381:
4378:
4374:
4366:
4363:
4360:
4356:
4353:
4349:
4346:
4335:
4332:
4320:
4317:
4314:
4311:
4308:
4305:
4297:
4272:
4269:
4266:
4263:
4260:
4257:
4249:
4224:
4221:
4218:
4189:
4186:
4183:
4163:
4160:
4155:
4151:
4147:
4142:
4138:
4134:
4129:
4125:
4121:
4118:
4115:
4112:
4107:
4103:
4099:
4094:
4090:
4086:
4081:
4077:
4073:
4070:
4067:
4064:
4061:
4041:
4021:
4016:
4012:
4008:
4003:
3999:
3995:
3990:
3986:
3982:
3979:
3976:
3971:
3967:
3963:
3958:
3954:
3950:
3945:
3941:
3937:
3934:
3931:
3911:
3908:
3905:
3878:
3875:
3872:
3852:
3847:
3844:
3841:
3837:
3833:
3825:
3822:
3817:
3813:
3809:
3784:
3781:
3778:
3770:
3767:
3762:
3758:
3754:
3738:
3737:
3726:
3723:
3720:
3717:
3714:
3709:
3705:
3701:
3696:
3692:
3688:
3685:
3682:
3679:
3663:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3617:
3603:
3598:
3594:
3590:
3587:
3584:
3581:
3550:
3518:
3513:
3509:
3505:
3502:
3499:
3496:
3493:
3490:
3482:
3457:
3437:
3417:
3397:
3394:
3391:
3388:
3385:
3380:
3375:
3371:
3367:
3362:
3358:
3354:
3349:
3344:
3340:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3291:
3252:
3249:
3248:
3247:
3244:
3241:
3227:
3222:
3218:
3214:
3211:
3191:
3188:
3185:
3151:
3148:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3065:
3044:
3041:
3037:
3034:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2976:
2973:
2970:
2967:
2964:
2961:
2950:
2949:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2899:
2879:
2876:
2873:
2870:
2867:
2864:
2842:
2838:
2834:
2831:
2820:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2770:
2767:
2764:
2761:
2758:
2755:
2730:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2671:continuous map
2649:
2645:
2641:
2637:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2610:
2606:
2602:
2597:
2593:
2570:
2566:
2562:
2557:
2553:
2549:
2544:
2540:
2519:
2490:
2485:
2481:
2477:
2474:
2471:
2468:
2443:
2440:
2437:
2434:
2431:
2428:
2424:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2397:
2393:
2364:
2340:
2335:
2331:
2327:
2324:
2321:
2318:
2310:
2307:
2304:
2301:
2273:
2269:
2265:
2262:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2211:, also called
2188:
2184:
2181:
2178:
2175:
2172:
2169:
2163:
2155:
2151:
2147:
2144:
2121:
2116:
2112:
2108:
2105:
2102:
2099:
2091:
2088:
2085:
2082:
2045:
2041:
2018:
2014:
2010:
2007:
1987:
1984:
1981:
1978:
1975:
1972:
1968:
1961:
1955:
1952:
1949:
1929:
1926:
1923:
1920:
1917:
1914:
1910:
1889:
1886:
1883:
1862:
1858:
1855:
1852:
1849:
1820:
1816:
1812:
1809:
1804:
1800:
1796:
1793:
1784:and such that
1771:
1767:
1740:
1736:
1708:
1705:
1700:
1696:
1669:
1646:
1641:
1637:
1633:
1628:
1624:
1620:
1617:
1614:
1594:
1591:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1537:
1534:
1529:
1525:
1521:
1518:
1498:
1476:
1472:
1451:
1439:it generates.
1426:
1422:
1398:
1373:
1370:
1367:
1364:
1361:
1358:
1354:
1350:
1347:
1344:
1341:
1338:
1335:
1331:
1326:
1322:
1293:
1290:
1287:
1284:
1281:
1278:
1274:
1270:
1267:
1264:
1261:
1258:
1255:
1251:
1248:
1221:
1218:
1215:
1212:
1209:
1206:
1180:
1175:
1171:
1164:
1156:
1152:
1148:
1118:
1113:
1109:
1102:
1094:
1090:
1086:
1060:
1056:
1052:
1048:
1043:
1039:
1035:
1030:
1026:
1004:
1000:
996:
992:
987:
983:
979:
974:
970:
948:
945:
941:
938:
935:
932:
921:
920:
898:
893:
889:
885:
880:
876:
872:
864:
848:
843:, also called
826:
822:
810:
792:
787:
783:
756:
752:
727:
701:
675:
671:
655:
652:
650:
647:
646:
645:
633:
630:
627:
624:
621:
618:
615:
592:
589:
586:
583:
580:
560:
557:
554:
551:
548:
537:
524:
521:
517:
514:
510:
507:
503:
500:
496:
493:
489:
486:
462:
459:
455:
451:
448:
444:
440:
437:
416:
413:
410:
407:
404:
393:
381:
378:
375:
372:
369:
366:
363:
335:
324:
312:
309:
306:
303:
300:
272:
248:
228:
201:
198:
195:
192:
189:
174:
171:
64:
61:
45:tensor product
9:
6:
4:
3:
2:
7049:
7038:
7035:
7033:
7030:
7029:
7027:
7012:
7004:
7002:
6994:
6992:
6984:
6983:
6980:
6966:
6963:
6961:
6958:
6956:
6952:
6948:
6944:
6942:
6940:
6933:
6931:
6928:
6926:
6923:
6922:
6920:
6917:
6913:
6903:
6900:
6897:
6893:
6890:
6889:
6887:
6885:
6877:
6871:
6868:
6866:
6863:
6861:
6858:
6856:
6855:Tetracategory
6853:
6851:
6848:
6845:
6844:pseudofunctor
6841:
6838:
6837:
6835:
6833:
6825:
6822:
6818:
6813:
6810:
6808:
6805:
6803:
6800:
6798:
6795:
6793:
6790:
6788:
6785:
6783:
6780:
6778:
6775:
6773:
6769:
6763:
6762:
6759:
6755:
6750:
6746:
6741:
6723:
6720:
6718:
6715:
6713:
6710:
6708:
6705:
6703:
6700:
6698:
6695:
6693:
6690:
6688:
6687:Free category
6685:
6684:
6682:
6678:
6671:
6670:Vector spaces
6667:
6664:
6661:
6657:
6654:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6628:
6626:
6624:
6620:
6610:
6607:
6605:
6602:
6598:
6595:
6594:
6593:
6590:
6588:
6585:
6583:
6580:
6579:
6577:
6575:
6571:
6565:
6564:Inverse limit
6562:
6560:
6557:
6553:
6550:
6549:
6548:
6545:
6543:
6540:
6538:
6535:
6534:
6532:
6530:
6526:
6523:
6521:
6517:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6490:Kan extension
6488:
6486:
6483:
6481:
6478:
6476:
6473:
6471:
6468:
6466:
6463:
6461:
6458:
6454:
6451:
6449:
6446:
6444:
6441:
6439:
6436:
6434:
6431:
6429:
6426:
6425:
6424:
6421:
6420:
6418:
6414:
6410:
6403:
6399:
6395:
6388:
6383:
6381:
6376:
6374:
6369:
6368:
6365:
6358:
6353:
6349:
6348:
6339:
6336:
6334:
6332:
6327:
6324:
6320:
6313:
6312:
6306:
6305:
6291:
6286:
6279:
6270:
6265:
6258:
6250:
6244:
6236:
6232:
6228:
6222:
6218:
6214:
6210:
6206:
6199:
6191:
6187:
6183:
6179:
6174:
6169:
6165:
6161:
6160:
6152:
6144:
6140:
6136:
6132:
6128:
6124:
6119:
6114:
6109:
6104:
6100:
6096:
6092:
6085:
6077:
6073:
6069:
6065:
6060:
6055:
6051:
6047:
6043:
6036:
6028:
6024:
6020:
6018:9781450355834
6014:
6010:
6006:
6002:
6001:
5993:
5984:
5979:
5972:
5964:
5960:
5953:
5939:
5935:
5928:
5920:
5916:
5912:
5905:
5897:
5893:
5886:
5878:
5874:
5867:
5859:
5855:
5851:
5847:
5843:
5836:
5822:
5818:
5812:
5804:
5800:
5797:(1): 55–112.
5796:
5792:
5791:
5783:
5770:
5766:
5762:
5756:
5752:
5748:
5744:
5740:
5735:
5730:
5726:
5722:
5715:
5707:
5703:
5699:
5692:
5685:(3): 185–205.
5684:
5680:
5673:
5669:
5659:
5655:
5651:
5650:vector spaces
5647:
5644:
5641:
5637:
5634:
5632:
5628:
5625:
5623:
5619:
5616:
5615:
5607:
5604:
5602:
5598:
5595:
5593:
5589:
5585:
5581:
5578:
5576:
5573:
5571:
5568:
5566:
5563:
5561:
5558:
5556:
5553:
5551:
5550:Consciousness
5548:
5546:
5543:
5541:
5538:
5536:
5533:
5531:
5528:
5527:
5526:
5515:
5512:
5509:
5505:
5502:
5499:
5498:knot diagrams
5495:
5492:
5489:
5485:
5482:
5479:
5475:
5472:
5471:
5470:
5469:
5459:
5457:
5453:
5449:
5445:
5441:
5432:
5427:
5421:
5414:
5412:
5411:post-selected
5408:
5404:
5400:
5396:
5392:
5388:
5383:
5381:
5377:
5372:
5370:
5366:
5342:
5338:
5334:
5328:
5322:
5319:
5313:
5310:
5296:
5287:
5264:
5260:
5256:
5254:
5246:
5243:
5237:
5231:
5225:
5216:
5212:
5208:
5206:
5198:
5192:
5189:
5183:
5180:
5166:
5165:
5164:
5152:
5140:
5128:
5119:
5105:
5099:
5096:
5093:
5090:
5070:
5067:
5061:
5058:
5055:
5048:
5015:
5012:
4975:
4972:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4873:
4858:
4846:
4837:
4831:
4827:
4811:
4805:
4802:
4799:
4793:
4790:
4787:
4779:
4776:
4772:
4768:
4752:
4746:
4743:
4740:
4732:
4729:
4728:
4727:
4725:
4721:
4717:
4707:
4705:
4701:
4697:
4694:
4690:
4686:
4681:
4662:
4659:
4647:
4644:
4638:
4631:
4628:
4624:
4618:
4606:
4601:
4598:
4594:
4588:
4574:
4564:
4561:
4549:
4546:
4538:
4534:
4529:
4511:
4508:
4496:
4493:
4487:
4480:
4477:
4473:
4467:
4456:
4452:
4449:
4445:
4442:
4416:
4413:
4409:
4403:
4389:
4379:
4376:
4364:
4361:
4358:
4354:
4351:
4347:
4344:
4336:
4333:
4318:
4315:
4306:
4270:
4267:
4258:
4219:
4203:
4202:
4201:
4187:
4184:
4181:
4158:
4153:
4149:
4145:
4140:
4136:
4132:
4127:
4123:
4116:
4110:
4105:
4101:
4097:
4092:
4088:
4084:
4079:
4075:
4068:
4065:
4062:
4059:
4039:
4014:
4010:
4006:
4001:
3997:
3993:
3988:
3984:
3977:
3969:
3965:
3961:
3956:
3952:
3948:
3943:
3939:
3932:
3929:
3922:of a diagram
3909:
3906:
3903:
3896:
3892:
3876:
3873:
3870:
3845:
3842:
3839:
3835:
3823:
3815:
3811:
3779:
3768:
3760:
3756:
3715:
3712:
3707:
3703:
3699:
3694:
3690:
3686:
3680:
3664:
3649:
3646:
3643:
3640:
3634:
3618:
3601:
3596:
3588:
3582:
3566:
3565:
3564:
3562:
3548:
3538:
3534:
3516:
3511:
3491:
3488:
3480:
3455:
3435:
3415:
3386:
3383:
3378:
3373:
3365:
3360:
3352:
3347:
3342:
3334:
3328:
3325:
3322:
3319:
3316:
3305:
3280:
3278:
3274:
3270:
3266:
3262:
3258:
3251:Combinatorial
3245:
3242:
3225:
3220:
3212:
3209:
3186:
3170:
3169:
3168:
3166:
3149:
3142:
3136:
3133:
3130:
3124:
3101:
3095:
3092:
3089:
3083:
3063:
3042:
3035:
3009:
3006:
3003:
2997:
2994:
2971:
2968:
2965:
2959:
2932:
2929:
2926:
2920:
2917:
2897:
2874:
2871:
2868:
2862:
2840:
2832:
2829:
2821:
2803:
2800:
2797:
2791:
2788:
2765:
2762:
2759:
2753:
2746:the image of
2745:
2744:
2743:
2724:
2718:
2715:
2712:
2700:
2697:
2694:
2688:
2682:
2679:
2672:
2668:
2663:
2647:
2631:
2625:
2622:
2619:
2613:
2608:
2600:
2595:
2591:
2568:
2555:
2547:
2542:
2538:
2509:
2504:
2488:
2483:
2475:
2438:
2435:
2432:
2418:
2412:
2409:
2406:
2400:
2395:
2382:
2377:
2375:
2362:
2353:
2338:
2333:
2325:
2319:
2308:
2302:
2271:
2263:
2260:
2237:
2234:
2231:
2222:
2214:
2210:
2205:
2203:
2182:
2176:
2173:
2170:
2161:
2153:
2145:
2142:
2134:
2119:
2114:
2106:
2089:
2065:
2061:
2043:
2016:
2008:
2005:
1982:
1979:
1976:
1970:
1959:
1950:
1947:
1924:
1921:
1918:
1912:
1887:
1884:
1881:
1856:
1853:
1850:
1847:
1839:
1834:
1818:
1810:
1807:
1802:
1794:
1769:
1756:
1738:
1726:
1723:and a set of
1722:
1703:
1698:
1686:
1683:
1660:
1639:
1631:
1626:
1618:
1605:, is a tuple
1604:
1600:
1590:
1573:
1567:
1558:
1555:
1535:
1523:
1519:
1516:
1496:
1470:
1440:
1420:
1412:
1388:
1329:
1324:
1320:
1312:
1308:
1249:
1246:
1239:
1234:
1178:
1173:
1169:
1162:
1154:
1150:
1146:
1116:
1111:
1107:
1100:
1092:
1088:
1084:
1058:
1054:
1041:
1033:
1028:
1024:
1002:
998:
985:
977:
972:
968:
946:
933:
930:
918:
914:
896:
891:
878:
870:
862:
849:
846:
842:
824:
811:
808:
790:
785:
772:
754:
741:
740:
739:
738:is given by:
718:
713:
699:
691:
673:
669:
661:
631:
625:
622:
619:
616:
613:
606:
590:
584:
581:
578:
558:
552:
549:
546:
538:
522:
519:
515:
508:
505:
501:
498:
494:
491:
487:
484:
477:
460:
457:
449:
446:
442:
438:
435:
414:
408:
405:
402:
394:
379:
373:
370:
364:
349:
333:
325:
310:
304:
301:
298:
290:
289:
288:
286:
270:
262:
246:
226:
219:
215:
199:
193:
190:
187:
180:
170:
168:
164:
163:Gottlob Frege
160:
156:
152:
148:
144:
140:
136:
132:
128:
124:
120:
116:
112:
108:
103:
101:
97:
93:
89:
85:
81:
77:
76:Roger Penrose
73:
69:
60:
58:
54:
50:
46:
42:
38:
37:vector spaces
34:
30:
26:
22:
18:
6935:
6916:Categorified
6820:n-categories
6806:
6771:Key concepts
6609:Direct limit
6592:Coequalizers
6510:Yoneda lemma
6416:Key concepts
6406:Key concepts
6330:
6310:
6278:
6257:
6208:
6198:
6163:
6157:
6151:
6098:
6094:
6084:
6049:
6045:
6035:
5999:
5992:
5971:
5962:
5952:
5942:, retrieved
5937:
5927:
5918:
5914:
5904:
5895:
5885:
5876:
5866:
5849:
5845:
5835:
5824:. Retrieved
5821:angg.twu.net
5820:
5811:
5794:
5788:
5782:
5772:, retrieved
5724:
5714:
5705:
5701:
5691:
5682:
5678:
5672:
5622:linear logic
5524:
5478:braid groups
5467:
5465:
5437:
5384:
5379:
5373:
5368:
5362:
5285:
5162:
4870:Consider an
4869:
4856:
4835:
4829:
4825:
4774:
4770:
4766:
4719:
4715:
4713:
4699:
4685:word problem
4682:
4537:interchanger
4536:
4530:
4433:
3894:
3890:
3739:
3540:
3303:
3281:
3277:generic form
3276:
3264:
3254:
3164:
2951:
2666:
2664:
2507:
2505:
2380:
2378:
2355:
2287:
2212:
2208:
2206:
2201:
2067:
2063:
2059:
1835:
1754:
1720:
1603:cell complex
1596:
1441:
1387:left adjoint
1311:free functor
1235:
922:
916:
912:
844:
840:
806:
770:
716:
714:
657:
284:
217:
178:
176:
138:
130:
104:
66:
29:2-categories
16:
15:
6884:-categories
6860:Kan complex
6850:Tricategory
6832:-categories
6722:Subcategory
6480:Exponential
6448:Preadditive
6443:Pre-abelian
6317:. Youtube.
6118:10230/53097
5654:linear maps
5592:ZX-calculus
5540:Game theory
2667:deformation
2209:progressive
2202:inner nodes
2200:are called
2060:outer nodes
1838:plane graph
1385:, i.e. the
1236:There is a
690:free monoid
688:denote the
660:Kleene star
605:composition
119:finite sets
92:Ross Street
88:André Joyal
68:Günter Hotz
41:linear maps
7026:Categories
6902:3-category
6892:2-category
6865:∞-groupoid
6840:Bicategory
6587:Coproducts
6547:Equalizers
6453:Bicategory
6290:1809.00738
6269:1804.07626
6173:1908.07021
6166:: 107239.
6101:: 103168.
5983:1711.10455
5944:2022-11-08
5826:2022-11-11
5774:2022-11-08
5708:: 221–244.
5664:References
5618:Proof nets
5395:Bell state
4872:adjunction
4032:by a type
3895:whiskering
3740:such that
3665:a list of
3408:of a type
2742:such that
649:Definition
6951:Symmetric
6896:2-functor
6636:Relations
6559:Pullbacks
6243:cite book
6190:201103837
6143:235683270
6127:1053-8100
6059:1102.2368
5769:115169297
5734:0903.0340
5329:η
5320:∘
5311:ε
5244:η
5238:∘
5232:ε
5199:η
5190:∘
5181:ε
5103:→
5091:ε
5065:→
5056:η
5026:→
4986:←
4902:ε
4896:η
4809:→
4797:⇒
4788:α
4780:a 2-cell
4750:→
4733:a 1-cell
4693:confluent
4648:⊗
4639:∘
4625:⊗
4595:⊗
4575:∘
4550:⊗
4497:⊗
4488:∘
4474:⊗
4446:⊗
4410:⊗
4390:∘
4365:⊗
4348:⊗
4185:⊗
4117:…
4063:⊗
3978:…
3907:⊗
3722:Σ
3713:∈
3700:…
3647:≥
3619:a length
3602:⋆
3593:Σ
3589:∈
3567:a domain
3517:⋆
3508:Σ
3504:→
3498:Σ
3393:Σ
3379:⋆
3370:Σ
3366:×
3357:Σ
3353:×
3348:⋆
3339:Σ
3335:∈
3290:Σ
3226:⋆
3217:Σ
3213:∈
3147:Γ
3131:−
3105:Γ
3090:−
3040:Γ
3036:∼
3033:Γ
2998:∈
2966:−
2921:∈
2837:Γ
2833:∈
2792:∈
2760:−
2725:×
2707:→
2689:×
2686:Γ
2644:Σ
2640:→
2632:×
2614:−
2605:Γ
2565:Σ
2561:→
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2518:Σ
2489:⋆
2480:Γ
2476:∈
2470:Γ
2427:→
2419:×
2401:−
2392:Γ
2339:⋆
2330:Γ
2326:∈
2268:Γ
2264:∈
2226:→
2213:recumbent
2183:×
2162:−
2150:Γ
2146:∈
2120:⋆
2111:Γ
2107:∈
2101:Γ
2084:Γ
2068:codomain
2040:Γ
2013:Γ
2009:∈
1971:×
1960:∩
1954:Γ
1951:∈
1913:×
1857:∈
1815:Γ
1811:∐
1799:Γ
1795:−
1792:Γ
1766:Γ
1735:Γ
1707:Γ
1704:⊆
1695:Γ
1668:Γ
1636:Γ
1623:Γ
1616:Γ
1593:Geometric
1565:→
1562:Σ
1533:→
1528:Σ
1475:Σ
1450:Σ
1425:Σ
1397:Σ
1353:→
1325:−
1273:→
1179:∘
1147:∘
1117:∘
1085:∘
1051:Σ
1047:→
1038:Σ
995:Σ
991:→
982:Σ
944:Σ
940:→
937:Σ
897:⋆
888:Σ
884:→
875:Σ
821:Σ
791:⋆
782:Σ
751:Σ
726:Σ
674:⋆
654:Algebraic
629:→
617:∘
588:→
556:→
513:→
488:⊗
454:→
412:→
377:→
308:→
291:each box
285:signature
214:processes
197:→
173:Intuition
125:with the
123:relations
43:with the
21:morphisms
7011:Glossary
6991:Category
6965:n-monoid
6918:concepts
6574:Colimits
6542:Products
6495:Morphism
6438:Concrete
6433:Additive
6423:Category
6319:Archived
6235:18492893
6135:34627099
6046:Synthese
6027:17887510
5612:See also
5601:DisCoCat
5452:DisCoCat
5397:and the
5045:and the
4861:Examples
4726:. Thus,
4722:, using
4663:′
4632:′
4602:′
4565:′
4512:′
4481:′
4453:′
4417:′
4380:′
4355:′
3863:for all
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2822:for all
1059:′
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947:′
917:codomain
658:Let the
603:, their
523:′
509:′
495:′
474:, their
461:′
450:′
439:′
348:identity
151:complete
100:PGF/TikZ
7001:Outline
6960:n-group
6925:2-group
6880:Strict
6870:∞-topos
6666:Modules
6604:Pushout
6552:Kernels
6485:Functor
6428:Abelian
6338:DisCoPy
6328:at the
6076:3736082
5739:Bibcode
2508:labeled
2381:generic
1753:called
1409:to the
261:systems
78:, with
63:History
6947:Traced
6930:2-ring
6660:Fields
6646:Groups
6641:Magmas
6529:Limits
6233:
6223:
6188:
6141:
6133:
6125:
6074:
6025:
6015:
5767:
5757:
5599:, see
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4965:where
4767:string
4642:
4636:
4578:
4572:
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4485:
4393:
4387:
3893:. The
3891:offset
3674:layers
3537:quiver
2165:
2159:
2064:domain
1963:
1957:
1682:closed
1166:
1160:
1104:
1098:
913:domain
812:a set
742:a set
476:tensor
346:, the
137:, the
129:. The
6941:-ring
6828:Weak
6812:Topos
6656:Rings
6285:arXiv
6264:arXiv
6231:S2CID
6186:S2CID
6168:arXiv
6139:S2CID
6072:S2CID
6054:arXiv
6023:S2CID
5978:arXiv
5765:S2CID
5729:arXiv
4235:with
3304:layer
3076:with
2855:, if
2463:boxes
1874:with
1755:edges
1721:nodes
1657:of a
845:boxes
807:types
218:wires
179:boxes
147:sound
96:LaTeX
6631:Sets
6249:link
6221:ISBN
6131:PMID
6123:ISSN
6013:ISBN
5755:ISBN
5656:and
5638:and
5586:and
5083:and
4941:and
4683:The
4283:and
3874:<
3795:and
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3263:, a
3259:and
3117:and
2066:and
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1680:, a
1134:and
1016:and
915:and
571:and
427:and
157:for
149:and
139:cuts
121:and
109:and
105:The
98:and
90:and
39:and
6475:End
6465:CCC
6333:Lab
6213:doi
6178:doi
6164:370
6113:hdl
6103:doi
6064:doi
6050:186
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5854:doi
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5799:doi
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5446:in
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