# 2-Dimensional Categories

@article{Johnson20212DimensionalC, title={2-Dimensional Categories}, author={Niles Johnson and Donald Yau}, journal={arXiv: Category Theory}, year={2021} }

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

- Mathematics
- 2022

. 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

- MathematicsArchive 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

- MathematicsTransactions 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

- Mathematics
- 2022

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

- MathematicsArXiv
- 2022

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

- Mathematics
- 2022

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 ScienceArXiv
- 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…