We will continue online on Thursday, February 18th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Charles Walker

Distributive laws, pseudodistributive laws and decagons

Abstract:
The notion of a distributive law of monads was introduced by Beck [1], and gives a concise description of the data required to compose monads. In the two dimensional case, Marmolejo [4] defined pseudodistributive laws of pseudomonads (where the required diagrams only commute up to an invertible modification). However, this description requires a number of coherence conditions due to the extra data involved.

In this talk we give alternative definitions of distributive laws and pseudodistributive laws involving the decagonal coherence conditions which naturally arise when the involved monads and pseudomonads are presented in their extensive form [7, 3, 2, 6]. As an application, we show that of Marmolejo and Wood’s eight coherence axioms for pseudodistributive laws [5], three are redundant.  We will then go on to give (likely) minimal definitions of distributive laws and pseudodistributive laws, which further simplify the coherence conditions involved in this extensive viewpoint.

References

[1] Jon Beck. Distributive laws. In Sem. on Triples and Categorical Homology Theory (ETH,
Zürich, 1966/67), pages 119–140. Springer, Berlin, 1969.

[2] M. Fiore, N. Gambino, M. Hyland, and G. Winskel. Relative pseudomonads, Kleisli bicate-
gories, and substitution monoidal structures. Selecta Math. (N.S.), 24(3):2791–2830, 2018.

[3] Ernest G. Manes. Algebraic theories. Springer-Verlag, New York-Heidelberg, 1976. Graduate
Texts in Mathematics, No. 26.

[4] F. Marmolejo. Distributive laws for pseudomonads. Theory Appl. Categ., 5:No. 5, 91–147,
1999.

[5] F. Marmolejo and R. J. Wood. Coherence for pseudodistributive laws revisited. Theory Appl.
Categ., 20:No. 5, 74–84, 2008.

[6] F. Marmolejo and R. J. Wood. No-iteration pseudomonads. Theory Appl. Categ., 28:No. 14,
371–402, 2013.

[7] Robert Frank Carslaw Walters. A categorical approach to universal algebra. PhD thesis, Australian National University, 1970.

Invitation to dissertation defense of Jakub Juranek Thursday, 18.2.2021, from 11:00 am

Please connect under your full name and surname.

Link: https://cesnet.zoom.us/j/97365408217

We will continue online on Thursday, February 11th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Paolo Perrone

Kan extensions are partial colimits

Abstract:
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", where one replaces parts of a diagram by their colimits. We make this intuition precise by means of the "partial evaluations" sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the category CAT of locally small categories. 

We also define a morphism of monads between them, which we call "image", and which takes the "free colimit" of a diagram. This morphism allows us in particular to generalize the idea of "confinal functors", i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case.
The main result of this work says that a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.

Joint work with Walter Tholen. arXiv:2101.04531.

We will continue online on Thursday, February 4th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Jiří Rosický

Metric monads

Abstract:
We develop universal algebra over an enriched category and relate it to finitary enriched monads. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.