## April 29, 2019

### Right Properness of Left Bousfield Localizations

#### Posted by Mike Shulman

(Guest post by Raffael Stenzel)

This post is a sequel to the discussion of the mysterious nature of right properness and its understanding as an instance of coherence problems for presenting $(\infty,1)$-categorical structure. The last post discussed a relation between right properness of a model category $\mathcal{M}$ and locally cartesian closedness of the underlying $(\infty,1)$-category $\mathrm{Ho}_{\infty}(\mathcal{M})$. While the two properties – that is right properness of $\mathcal{M}$ on the one hand and locally cartesian closedness of $\mathrm{Ho}_{\infty}(\mathcal{M})$ on the other – are generally independent of each other, the post and its subsequent discussion basically established an equivalence of the two properties in the context of Cisinski model categories in the following sense; a presentable $(\infty,1)$-category $\mathcal{C}$ is locally cartesian closed iff there is a right proper Cisinski model category $\mathcal{M}$ whose underlying $(\infty,1)$-category is equivalent to $\mathcal{C}$. In this follow up, we aim to generalize this connection, and we do so via replacing “locally cartesian closedness” of $(\infty,1)$-categories by “semi-left exactness” of their reflective localizations.

While this is not meant to be an exhaustive description of the nature of right properness either, it hopefully gives another stimulus to kindle further discussion.

Posted at 7:43 PM UTC | Permalink | Followups (16)

## April 28, 2019

### Generalized Petri Nets

#### Posted by John Baez

I just finished a paper which uses Lawvere theories to generalize Petri nets. I can think of two reasons why people might be interested in this:

• Category theorists love Lawvere theories and are in awe of their power. However, it can be hard to find instances where Lawvere theories are used to get something specific and practical accomplished.

• There are lots of papers on Petri nets and their variants. The bibliography on Petri nets world has over 8500 citations. This generalization puts some of the more popular variants under a common framework and allows for exploration of the relationships between them.

Posted at 3:36 AM UTC | Permalink | Followups (12)

## April 24, 2019

### Twisted Cohomotopy Implies M-Theory Anomaly Cancellation

#### Posted by David Corfield

The latest instalment of Urs’s march towards M-theory is out on the arXiv today, Twisted Cohomotopy implies M-Theory anomaly cancellation, (arXiv:1904.10207).

A review of the program to date appears in the recent:

In the latest paper the authors are moving beyond the approximation of rational cohomology to explain from first principles the folklore about anomaly cancellations in M-theory. The generalized non-abelian cohomology theory known as cohomotopy theory, and in particular its twisted version, appears to have all the answers, accounting for six conditions on the cocycles corresponding to the C-field.

Posted at 11:11 AM UTC | Permalink | Followups (13)

## April 19, 2019

### Can 1+1 Have More Than Two Points?

#### Posted by John Baez

I feel I’ve asked this before… but now I really want to know. Christian Williams and I are working on cartesian closed categories, and this is a gaping hole in my knowledge.

Question 1. Is there a cartesian closed category with finite coproducts such that there exist more than two morphisms from $1$ to $1 + 1$?

Cartesian closed categories with finite coproducts are a nice context for ‘categorified arithmetic’, since they have $0$, $1$, addition, multiplication and exponentiation. The example we all know is the category of finite sets. But every cartesian closed category with finite coproducts obeys what Tarski called the ‘high school algebra’ axioms:

$x + y \cong y + x$

$(x + y) + z \cong x + (y + z)$

$x \times 1 \cong x$

$x \times y \cong y \times x$

$(x \times y) \times z \cong x \times (y \times z)$

$x \times (y + z) \cong x \times y + x \times z$

$1^x \cong 1$

$x^1 \cong x$

$x^{(y + z)} \cong x y \times x z$

$(x \times y)^z \cong x^z \times y^z$

$(x^y)^z \cong x^{(y \times z)}$

together with some axioms involving $0$ which for some reason Tarski omitted: perhaps he was scared to admit that in this game we want $0^0 = 1$.

So, one way to think about my question is: how weird can such a category be?

Posted at 4:43 AM UTC | Permalink | Followups (35)

## April 17, 2019

### Univalence in (∞,1)-toposes

#### Posted by Mike Shulman

It’s been believed for a long time that homotopy type theory should be an “internal logic” for $(\infty,1)$-toposes, in the same way that higher-order logic (or extensional type theory) is for 1-toposes. Over the past decade a lot of steps have been taken towards realizing this vision, but one important gap that’s remained is the construction of sufficiently strict universe objects in model categories presenting $(\infty,1)$-toposes: the object classifiers of an $(\infty,1)$-topos correspond directly only to a kind of “weakly Tarski” universe in type theory, which would probably be tedious to use in practice (no one has ever seriously tried).

Yesterday I posted a preprint that closes this gap (in the cases of most interest), by constructing strict univalent universe objects in a class of model categories that suffice to present all Grothendieck $(\infty,1)$-toposes. The model categories are, perhaps not very surprisingly, left exact localizations of injective model structures on simplicial presheaves, which were previously known to model all the rest of type theory; the main novelty is a new more explicit “algebraic” characterization of the injective fibrations, enabling the construction of universes.

Posted at 12:35 AM UTC | Permalink | Followups (2)

## April 14, 2019

### The ZX-Calculus for Stabilizer Quantum Mechanics

#### Posted by John Baez

guest post by Fatimah Ahmadi and John van de Wetering

This is the second post of Applied Category Theory School 2019. We present Backens’ completeness proof for the ZX-calculus for stabilizer quantum mechanics.

Posted at 1:20 AM UTC | Permalink | Followups (3)

## April 9, 2019

### Postdoctoral Researcher Position in Lisbon

#### Posted by John Huerta

Applications are invited for a postdoctoral researcher position in the “Higher Structures and Applications” research team, funded by the Portuguese funding body FCT.

## April 4, 2019

### Category Theory 2019

#### Posted by Tom Leinster

I’ve announced this before, but registration is now open! Here we go…

Third announcement and call for contributions

Category Theory 2019

University of Edinburgh, 7–13 July 2019

Invited speakers:

plus an invited tutorial lecture on graphical linear algebra by

and a public event on Inclusion-exclusion in mathematics and beyond by