-
Notifications
You must be signed in to change notification settings - Fork 0
/
436sentences.txt
433 lines (433 loc) · 31.3 KB
/
436sentences.txt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
%%% Examples of sentences from TAC abstracts, the right size, no latex. 433/3188 total sentences, misnomed as 436Experiment
This yields a quadratic algorithm deciding the equality of diagrams in a free double category.
The right adjoint of this Quillen equivalence is the classical Segal's Nerve functor.
Along the way we prove numerous results showing that the enchilada category is rather strange.
Let PreOrd(C) be the category of internal preorders in an exact category C.
Persistence has proved to be a valuable tool to analyze real world data robustly.
This allows a definition of topological complexity for orbifolds.
This note shows every Grothendieck topos has such a site.
On the other hand subcanonical one-way sites are very special.
A site criterion for petit toposes will probably require subcanonical sites.
We call such 2-monads property-like.
We show that associative coequalizing multiplications suffice and call the resulting structures interpolads.
This property then extends to strong Colimits of sketches.
Often used implicitly, the precise statement of this property and its proof appears here.
However, we provide examples to show that the reflector and coreflector need not coincide.
Results on the finiteness of induced crossed modules are proved both algebraically and topologically.
Strong promonoidal functors are defined.
A construction for the free monoidal category on a promonoidal category is provided.
In this paper we study the lattice of quantic conuclei for orthomudular lattices.
We discuss two versions of a conjecture attributed to M. Barr.
We introduce MD-sketches, which are a particular kind of Finite Sum sketches.
Two interesting results about MD-sketches are proved.
As a corollary, we obtain that equivalence of data-specifications is decidable.
Their universal properties can then be derived with standard techniques as used in duality theory.
When it does, we call the string distributive.
These provide a new construction of the simplicial 2-category, Delta.
Here we show an analogous description of locally finitely multipresentable categories.
In addition, we define and study the appropriate categorical structure underlying the MIX rule.
We give the definition in any Gray-category.
The concept of algebra is given as an adjunction with invertible counit.
We show that these doctrines are instances of more general pseudomonads.
2-crossed complexes are introduced and similar freeness results for these are discussed.
Partial results are given for higher dimensions.
Applications to 2-crossed modules and quadratic modules are discussed.
The theory of enriched accessible categories over a suitable base category V is developed.
A particular attention is devoted to enriched locally presentable categories and enriched functors.
Consequently, an equivariant simplicial version of the Whitehead Theorem is derived.
Some examples are parity c omplexes, pasting schemes and directed complexes.
This role makes the relationship between projective objects and the tensor product especially critical.
Conditions are given under which such a structure interacts appropriately with projective objects.
These examples were not fabricated to illustrate the abstract possibility of misbehavior.
Rather, they are drawn from the literature.
We give an abstract characterization of categories which are localizations of Maltsev varieties.
These results can be applied to characterize localizations of naturally Maltsev varieties.
Using the Chu-construction, we define a group algebra for topological Hausdorff groups.
Exponentiable spaces are characterized in terms of convergence.
These isomorphisms must satisfy nine coherence conditions.
We define what is a pseudomonad with compatible structure with respect to two given pseudomonads.
This all extends routinely to local presentability with respect to any regular cardinal.
A number of examples are considered.
The main tool in the new approach is the Chu construction.
Finally, we show loop and suspension functors in the pointed case.
The purpose of this paper is to indicate some bicategorical properties of ring theory.
In this interaction, static modules are analyzed.
This leads to the notion of strongly protomodular category.
The appendix provides the definitions of a braided monoidal bicategory and sylleptic monoidal bicategory.
An application to exact sequences is given.
In this paper we show that under slightly stronger assumptions the converse is also true.
We present some new findings concerning branched covers in topos theory.
Finally, we characterize when certain categories of sheaves are toposes.
We present two other characterizations.
A relationship between pseudoepimorphisms and lax epimorphisms is discussed.
The construction is performed in two steps.
First a cartesian closed extension L of CLS is obtained.
We only treat the branching side.
The 2-category VAR of finitary varieties is not varietal over CAT.
An equational hull of VAR w.r.t. all operations is also discussed.
We give a self-contained presentation of Batanin's construction that suits our purposes.
An important example: reflexive coequalizers are sifted colimits.
Among complete categories, generalized varieties are precisely the varieties.
A feature of a ramification groupoid is that it carries a certain order structure.
Our work extends naturally to the braid group on countably many generators.
We show that the two notions are equivalent.
This is simultaneously an extension of Verdier's version of Cech cohomology to homotopy.
Exact sequences are a well known notion in homological algebra.
As an application, we compare presheaf categories and varieties.
The monadic arrows Opmon are then characterized.
We call such structures entropic categories.
We demonstrate the soundness and completeness of our axiomatization with respect to cut-elimination.
We then focus on several methods of building entropic categories.
Several examples are discussed, based first on the notion of a bigroup.
Finally the Tannaka-Krein reconstruction theorem is extended to the entropic setting.
The symbolic and categorical structures are thereby shown to be equivalent.
We obtain a characterization of such functors.
A cyclic spectrum is constructed for Boolean flows.
Examples include attractive fixpoints, repulsive fixpoints, strange attractors and the logistic equation.
The primitive symbolic and categorical structures are extended to make their types sober.
A more technical characterization of axiomatizable classes in geometric logic is presented.
We restrict our study to the case of locally partially ordered bases.
We give two applications: sheaves over locales and group actions.
Hopf formulas for the second and third homology of a Lie algebra are proved.
The construction is based on the geometric notion of thin square.
The definitions can be read independently.
It is shown that every codescent morphism of groups is effective.
We generalize Dress and Müller's main result in Decomposable functors and the exponential principle.
Subsets with this property are called composition-representative.
This paper studies lax higher dimensional structure over bicategories.
The general notion of a module between two morphisms of bicategories is described.
The composite of two such modules need not exist.
These modules and their modulations then give rise to a bicategory.
We give an explicit construction of the category Opetope of opetopes.
This result encompasses many known and new examples of quasitopoi.
We take some first steps in providing a synthetic theory of distributions.
We introduce various notions of partial topos, i.e. `topos without terminal object'.
Examples for the weaker notions are local homeomorphisms and discrete fibrations.
In such a framework, the globular nerve always satisfies the Kan condition.
We give a categorical discussion of such results.
Another is to make clear which parts of the proofs of such results are formal.
In this case we recover the notion of a linear bicategory.
The poly notions of functors, modules and their transformations are introduced as well.
These in fact correspond to modules having special properties.
In many applications of quasigroups isotopies and homotopies are more important than isomorphisms and homomorphisms.
We prove various results relating it to exponentiation of locales, including the following.
Those classes are natural examples of reflective subcategories defined by proper classes of morphisms.
Adamek and Sousa's result follows from ours.
The paper develops the previously proposed approach to constructing factorization systems in general categories.
The problem of relating a factorization system to a pointed endofunctor is considered.
Some relevant examples in concrete categories are given.
The symmetric case can easily be recovered.
This paper proposes a recursive definition of V-n-categories and their morphisms.
Our result relies heavily on some unpublished work of A. Kock from 1989.
The required simplicial approximation results for simplicial sets and their proofs are given in full.
Subdivision behaves like a covering in the context of the techniques displayed here.
Several exact sequences, relative to a subfunctor of the identity functor, are obtained.
The resulting notion of centrality fits into Janelidze and Kelly's theory of central extensions.
The centre of a monoidal category is a braided monoidal category.
Monoidal categories are monoidal objects (or pseudomonoids) in the monoidal bicategory of categories.
Some properties and sufficient conditions for existence of the construction are examined.
Having many corollaries, this was an extremely useful result.
Moreover, as the authors soon suspected, it specializes a much more general result.
We characterize semi-abelian monadic categories and their localizations.
We add some guiding examples.
We generalize to an arbitrary variety the von Neumann axiom for a ring.
We study its implications on the purity of monomorphisms and the flatness of algebras.
However, we also present a non-varietor satisfying Birkhoff's Variety Theorem.
It turns out that many categorical properties are well behaved under enlargements.
We describe a completion of gms's by Cauchy filters of formal balls.
The completion generalizes the usual one for metric spaces.
Various examples and constructions are given, including finite products.
A precise concept of concrete geometrical category is introduced in an axiomatic way.
In this article, we present a general construction of such an extension.
These results have recently been generalised to all dimensions by Philip Higgins.
We give two related universal properties of the span construction.
The first involves sinister morphisms out of the base category and sinister transformations.
There are no coherence requirements.
We describe this semantics incrementally.
Then we look at the universal property required to interpret each type constructor.
And we give ways of constructing models from other models.
This conjecture is false, but there is an equational characterization of absolute homology.
The main theorem comes close to a characterisation of this phenomenon.
This work is a contribution to a recent field, Directed Algebraic Topology.
This paper constructs models of intuitionistic set theory in suitable categories.
We provide various examples of this situation of a combinatorial nature.
For the elementary case, little more is known.
Finally, models of these theories are constructed in the category of ideals.
We discuss holonomy and prove an analogue of the Ambrose-Singer theorem.
The main result is now, that such monads form a stack.
The equivalence is FOLDS equivalence of the FOLDS-Specifications of the two concepts.
A representation theory for (strict) categorical groups is constructed.
Each categorical group determines a monoidal bicategory of representations.
Typically, these bicategories contain representations which are indecomposable but not irreducible.
A simple example is computed in explicit detail.
Universal properties of these constructions are presented.
We then compute the cyclic spectrum of any finitely generated Boolean flow.
Characterization theorems for unitary restriction categories are derived.
Let L be an arbitrary orthomodular lattice.
The proof of the thin filler conditions uses chain complexes and chain homotopies.
We give a Dialectica-style interpretation of first-order classical affine logic.
The constructions have the same objects, but are rather different in other ways.
We also point out interesting open problems concerning the Dialectica construction.
This view helps to clarify the composition of Chu-spans.
Here we introduce collarable (and collared) cospans between topological spaces.
Nuclei of categories of modules are considered as an example.
We then use these equations to classify operads with coherent unit actions.
We discuss an approach to constructing a weak n-category of cobordisms.
For general dimensions k and n we indicate what the construction should be.
We give here a positive answer to this question.
A question was left open: is there more structure yet to be defined?
In the process a number of facts about abstract core algebras must be developed.
The mistaken version is used later in that paper.
The straightforward extension to crossed complexes is also considered.
Bicat is the tricategory of bicategories, homomorphisms, pseudonatural transformations, and modifications.
We show that these two tricategories are not triequivalent.
It is shown that a cartesian bicategory is a symmetric monoidal bicategory.
Those systems that have a unique solution in every iterative algebra are characterized.
Some consequences and applications are presented.
We then describe two ways to construct framed bicategories.
In particular, the theory of Kan extensions extends to the setting of Grothendieck derivators.
In particular, Rep(C, D) always has an initial object.
In those two cases, the functors in question may have surprisingly opulent structures.
The latter leads to a deeper understanding of the notion of linear functor.
We replace connected components by constructively complemented, or definable, monomorphisms.
This paper deals with Kan extensions in a weak double category.
We show that their use can be avoided and all remaining results remain correct.
See note on p. 24.
Our paper extends these ideas somewhat.
We give some examples in algebra and in topos theory.
This paper revisits the authors' notion of a differential category from a different perspective.
We present an analysis of the basic properties of Cartesian differential categories.
The retract (A, a) is not free in general.
We study convergent (terminating and confluent) presentations of n-categories.
We characterise this property by using the notion of critical branching.
This paper introduces the notions of vector field and flow on a general differentiable stack.
Both of them generalise the concept of algebra on a monad T.
We define eventually cyclic Boolean flows and the eventually cyclic spectrum of a Boolean flow.
We axiomatically define (pre-)Hilbert categories.
The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor.
Neither enrichment nor a complex base field is presupposed.
A comparison to other approaches will be made in the introduction.
Distributive laws between monads (triples) were defined by Jon Beck in the 1960s.
Particular cases are the entwining operators between algebras and coalgebras.
For such a class of spaces homotopy orthogonality implies enriched orthogonality.
The state space of a machine admits the structure of time.
The set of such components often gives a computable invariant of machine behavior.
In the general case, no such meaningful partition could exist.
Thus we hope to extend geometric techniques in static program analysis to looping processes.
The present paper starts by supplying this last clause with a precise meaning.
This framework allows us to define push-forwards for Witt groups, for example.
Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma.
Under condition (I), every multiplicative graph is an internal category.
We describe a simplified categorical approach to Galois descent theory.
We conclude with applications to examples.
Now coproduct preservation yields an approach to product measures.
Then we present three applications of groupoidification.
The first is to Feynman diagrams.
The second application is to Hecke algebras.
The third application is to Hall algebras.
We present two generalizations of the Span construction.
Let R be a commutative ring whose complete ring of quotients is R-injective.
This allows presentation of the stochastic automata as algebras for this distributive law.
The ``if" directions fail for semi-abelian varieties.
We compute some simple examples and explore the elementary properties of these invariants.
The result then applies to quantum categories and bialgebroids.
But, is this really the correct level of generalisation?
Symbolic dynamics is partly the study of walks in a directed graph.
We determine the resulting homotopy category.
To each graph we associate a basal graph, well defined up to isomorphism.
We combine two recent ideas: cartesian differential categories, and restriction categories.
The category of Set-valued presheaves on a small category B is a topos.
A flow on a compact Hausdorff space is an automorphism.
Vertical arrows give rise to modules between representables.
Various concerns suggest looking for internal co-categories in categories with strong logical structure.
We give a new proof of the fact that every topos is adhesive.
We compare various different definitions of "the category of smooth objects".
This therefore applies beyond the question of categories of smooth spaces.
We indicate also some possible novel geometric interest in such algebras.
In this paper we will give a new, elementary proof of this result.
We clarify details of that work.
Our main conceptual tool is a monad on the category of grouped toposes.
We also discuss some new examples and results motivated by this characterization.
It also gives an easy way to calculate the sources and targets of opetopes.
Finally, we exhibit several free constructions relating the different classes of categories under consideration.
Our main results are presented at the level of monoidal bicategories.
In fact, we provide some very general results connecting opmonoidal monads and skew monoidales.
We prove that congruence lattices of Morita equivalent small categories are isomorphic.
In the final section applications to cartesian monoidal categories are considered.
We develop an alternative approach to star-autonomous comonads via linearly distributive categories.
Further applications are given.
The paper presents algebraic and logical developments.
This development involves a study of clone and double-dualization structures.
We also prove that each construction preserves comprehension.
We study the composition of modules between lax functors of weak double categories.
These axioms are commutative squares involving only co-smash products.
In this paper we introduce the corresponding morphism concept and examine its properties.
Moreover, we show that separated L-complete morphisms belong to a factorization system.
In fact, even Higgins commutators suffice.
When κ=ω it includes the pretopos completion of a coherent category.
We prove four characterization theorems dealing with adhesive categories and their variants.
There is a lot of redundancy in the usual definition of adjoint functors.
We define and prove the core of what is required.
First we do this in the hom-enriched context.
Finally, we describe a doctrinal setting.
Such a theory is called a super Fermat theory.
Any category of superspaces and smooth functions has an associated such theory.
This includes both real and complex supermanifolds, as well as algebraic superschemes.
These include conditions on relations as well as conditions on simplicial objects.
We also give various examples and counterexamples.
Cartesian differential categories abstractly capture the notion of a differentiation operation.
We show that this exterior derivative, as expected, produces a cochain complex.
It follows that complicial sets are equivalent to sets with complicial identities.
The resulting structures are called sets with complicial identities.
These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory.
We also briefly study 3-categories with weak inverses.
We finish by two questions about the problem suggested by the title of this text.
This proposition is then applied to the study of quotient geometric theories and subtoposes.
We introduce an axiomatic framework for the parallel transport of connections on gerbes.
Suppose that S is a space.
This result is analogous to the correspondence between measures and integrals.
So, for example, we can speak of integration against an ultrafilter.
We are particularly interested in pseudomonads that arise from KZ-doctrines.
Tightly bounded KZ-doctrines are shown to be idempotent.
We characterize topos inclusions corresponding to a general form of relative computability.
We characterize pcas whose realizability topos admits a geometric morphism to the effective topos.
We thereby resolve two of the problems with known approaches to bicategorical localization.
We show that the answer is positive by building some examples.
We set forth the basic theory of such enriched factorization systems.
The possibility of spectra in other categories is discussed.
We introduce a category that represents varying risk as well as ambiguity.
Then, we reformulate dynamic monetary value measures as a presheaf for the category.
We use a number of interesting categories related to probability theory.
A notion of central importance in categorical topology is that of topological functor.
In the nilpotent case, this nerve is known to be a Kan complex.
We work through numerous examples to demonstrate the power of these notions.
If the category is additive, we define a sheaf of categories of analytic functions.
We use these relations to define analytic versions of Arakelov compactifications of affine arithmetic varieties.
They are used in the paper only to prove Corollary~8.3.
A proof of this corollary is given without them.
Here ``balanced'' can be omitted if the category is additive.
We introduce the notion of mutation pairs in pseudo-triangulated categories.
This result unifies many previous constructions of quotient triangulated categories.
We study extra assumptions on pretopologies that are needed for this theory.
We check these extra assumptions in several categories with pretopologies.
Functors between groupoids may be localised at equivalences in two ways.
We show that both approaches give equivalent bicategories.
In this paper, we use the language of operads to study open dynamical systems.
The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams.
Moreover it enables us to characterise operads as categorical polynomial monads in a canonical way.
have two useful gradings related by isomorphisms which change the degree.
The result is a double category C//G which describes the local symmetries of C.
There are few known computable examples of non-abelian surface holonomy.
Using these ideas, we also prove that magnetic monopoles form an abelian group.
We introduce a 3-dimensional categorical structure which we call intercategory.
We show that these fit together to produce a strict triple category of intercategories.
This is the third paper in a series.
The effect of any bundle of Lie groups is trivial.
All quotients of a given Lie groupoid determine the same effect.
Our analysis is relevant to the presentation theory of proper smooth stacks.
This paper extends the Day Reflection Theorem to skew monoidal categories.
Let C be a finite category.
We also give a presentation for FinRelk.
For full subcategories the answer is affirmative.
We put a model structure on the category of categories internal to simplicial sets.
We also study the homotopy theory of internal presheaves over an internal category.
Monoidal differential categories provide the framework for categorical models of differential linear logic.
The coKleisli category of any monoidal differential category is always a Cartesian differential category.
We focus primarily on the definitions due to Batanin and Leinster.
In both cases we define the join operation as union.
In the present article we describe constructions of model structures on general bicomplete categories.
Constant cosheaves are constructed, and there are established connections with shape theory.
Unlike the uniform completion, the Dedekind completion of a vector lattice is not functorial.
Our examples include the gleaf of metric spaces and the gleaf of joint probability distributions.
Examples of compositories include nerves of categories and compositories of higher spans.
We also characterize the irreducible and indecomposable representations.
We consider locales B as algebras in the tensor category sl of sup-lattices.
This equivalence follows from two independent results.
We do not require the base monoidal category M to be closed or symmetric monoidal.
We study the properties of these functors, and calculate some examples.
We give several examples of compact closed bicategories, then review previous work.
We develop a homotopy theory of categories enriched in a monoidal model category V.
The model structure is shown to be left proper.
Our results hold in any pointed regular protomodular category.
We show that the morphism axiom for n-angulated categories is redundant.
In this situation we need the notion of a delocalization.
Here we establish its significance in an algebraic context.
Birkhoff's variety theorem from universal algebra characterises equational subcategories of varieties.
We give an analogue of Birkhoff's theorem in the setting of enrichment in categories.
We will also see that these constructions preserve right properness and compatibility with simplicial enrichment.
The purpose of this paper is two-fold.
In the case of sheaves, we use local equivalence as the weak equivalence.
The theory of derivators enhances and simplifies the theory of triangulated categories.
We present a theory of cohomological as well as homological descent in this language.
The main motivation is a descent theory for Grothendieck's six operations.
Several consequences of examining particular classes of hypercrossed complexes of Lie algebras are presented.
Two states are past-similar if they have homotopic pasts.
However, R(S) can be seen as a new type of quantale completions of S.
Moreover, we exhibit the precise relationship between holomorphic and smooth gerbes.
This is phrased in terms of a multilinear functor calculus in a bicategory.
Connections are an important tool of differential geometry.
This paper investigates their definition and structure in the abstract setting of tangent categories.
A compactification technique is developed based on Shanin's method.
Among these are different versions of the acyclic models method.
Such a functor between effective categories is known to be an equivalence.
Databases have been studied category-theoretically for decades.
The aim of this paper is to present a perspective that addresses this issue.
Throughout, we emphasise how corelations model interconnection.
All these structures can be realised as algebras over polynomial monads.
Many important monads are shown to be tame polynomial.
This construction generalizes the nerve construction for differential graded categories given by Lurie.
We study the structure of the category of polynomials in a locally cartesian closed category.
This composition, however, is somewhat complex and difficult to work with.
Our first important result is similar to that of Lack and Street.
We study deformation of tube algebra under twisting of graded monoidal categories.
Let H be a quasi-Hopf algebra.
We prove that this functor is symmetric monoidal and indeed a hypergraph functor.
Each kind of circuit corresponds to a mathematically natural prop.
We describe the `behavior' of these circuits using morphisms between props.
We introduce and compare several new exactness conditions involving what we call split cubes.
The proof uses only certain categorical properties of the category of locales, Loc.
This 2-categorical refinement also provides a uniqueness statement concerning canonical triangulations.
We introduce a new class of categories generalizing locally presentable ones.
These biequivalences induce equivalences between the corresponding categories of algebras.
There are three main ingredients in establishing these biequivalences.
One motivation for this construction is an application to graph rewriting.
In this paper, we study properties of maps between fibrant objects in model categories.
We give a characterization of weak equivalences between fibrant objects.
An open question was: is it also a variety?
We show that the answer is affirmative.
The dual equivalence is induced by the dualizing object [0,1].
These variants of universal algebra are called {notions of algebraic theory}.
In this paper, we develop a unified framework for them.
In this first installment we introduce the free globularly generated double category construction.
Some concrete examples of categories where these results hold are examined in detail.
First, we characterize which prederivators arise on the nose as prederivators associated to quasicategories.
We also discuss the question when that last assumption can be dropped.
Furthermore, terminal coalgebras are often constructed as limits of a certain type.
We then give a syntactical characterization of left coextensive varieties of universal algebras.
We consider such a distributor a weak analogue of adjunction.
Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories.
We construct model category structures on various types of (marked) *-categories.
Let E be a topos.
Let level 1 be the Aufhebung of level 0.
We also discuss dual constructions.
Consequently it is a monoidal category and a Lie groupoid.
Among algebraic majority categories are the categories of lattices, Boolean algebras and Heyting algebras.
In addition, we introduce the notion of maximal rigid subcategories in extriangulated categories.
The case E = Set is of special interest.
We refer to this problem as the problem of existence of internalizations for decorated bicategories.
Motivated by this we introduce the condition of a double category being globularily generated.
This construction gives us a context of non-associative relative algebraic geometry.
The most important example of the construction is the octonionic projective space.
In this paper we develop a theory of Segal enriched categories.
We introduce Segal dg-categories which did not exist so far.
We analyze some of the sheaf theoretic aspects of this topos.
In this paper we prove an equivalence theorem originally observed by Robert MacPherson.
Analogous results for the associated homotopy limit (and other intermediate limits) directly follow.
We also construct semi-simplicial versions of all these.
Restriction categories were introduced as a way of generalising the notion of partial map category.
Cheng, Gurski, and Riehl constructed a cyclic double multicategory of multivariable adjunctions.
We introduce `network models' to encode these ways of combining networks.
Different network models describe different kinds of networks.
Such operads, and their algebras, can serve as tools for designing networks.
Necessary and sufficient conditions for the implication are presented.
Examples are shown to arise from 2-category theory and from bialgebras.