Další seminář z algebry se koná 18.2.2021 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Charles Walker

Distributive laws, pseudodistributive laws and decagons

Abstrakt:
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.

Další seminář z algebry se koná 11.2.2021 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Paolo Perrone

Kan extensions are partial colimits

Abstrakt:
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.