# 2-Dimensional Categories

@article{Johnson20212DimensionalC,
title={2-Dimensional Categories},
author={Niles Johnson and Donald Yau},
journal={arXiv: Category Theory},
year={2021}
}
• Published 14 February 2020
• Mathematics
• arXiv: Category Theory
2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the…
68 Citations

### Extensions of representation stable categories

. A category of FI type is one which is suﬃciently similar to ﬁnite sets and injections so as to admit nice representation stability results. Several common examples admit a Grothendieck ﬁbration to

### Type space functors and interpretations in positive logic

• M. Kamsma
• Mathematics
Archive for Mathematical Logic
• 2022
We construct a 2-equivalence $$\mathfrak {CohTheory}^{op }\simeq \mathfrak {TypeSpaceFunc}$$ CohTheory op ≃ TypeSpaceFunc . Here $$\mathfrak {CohTheory}$$ CohTheory is the

### An (∞,2)-categorical pasting theorem

• Mathematics
Transactions of the American Mathematical Society
• 2022
We show that any pasting diagram in any ( ∞ , 2 ) (\infty ,2) -category has a homotopically unique composite. This is achieved by showing that the free 2-category generated by a

### Notes on Lax Ends

In enriched category theory, the notion of extranatural transformations is more fundamental than that of ordinary natural transformations, and the ends, the universal extranatural transformations,

### Space-time tradeoffs of lenses and optics via higher category theory

Optics and lenses are abstract categorical gadgets that model systems with bidirectional data ﬂow. In this paper we observe that the denotational deﬁnition of optics – identifying two optics as

### Rigid models for 2-gerbes I: Chern-Simons geometry

• Mathematics
• 2022
Motivated by the problem of constructing explicit geometric string structures, we give a rigid model for bundle 2-gerbes, and define connective structures thereon. This model is designed to make

### 2-Representations of Lie 2-groups and 2-Vector Bundles

Murray, Roberts and Wockel showed that there is no strict model of the string 2group using the free loop group. Instead, they construct the next best thing, a coherent model for the string 2-group

### Bicategories of Action Groupoids

• Mathematics
• 2022
. We prove that the 2-category of action Lie groupoids localised in the following three diﬀerent ways yield equivalent bicategories: localising at equivariant weak equivalences à la Pronk, localising

### K-theoretic classification of inductive limit actions of fusion categories on AF-algebras

• Mathematics
• 2022
We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C∗-algebras. We show that for inductive limits of finite dimensional actions of fusion categories on

### Structured Decompositions: Structural and Algorithmic Compositionality

• Mathematics, Computer Science
ArXiv
• 2022
This work proves an algorithmic meta theorem for theSubP -COMPOSITION problem which, when instantiated in the category of graphs, yields compositional algorithms for NP-hard problems such as: MAXIMUM BIPARTITE SUBGRAPH, MAXIMum PLANAR SUB GRAPH and LONGEST PATH.

## References

SHOWING 1-10 OF 143 REFERENCES

### Constructing symmetric monoidal bicategories functorially.

• Mathematics
• 2019
We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many