The λ-calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses βη-conversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from values to values ) that may jeopardise the applicability of theoretical results. In this paper we introduce calculi, based on a categorical semantics for computations , that provide a correct basis for proving equivalence of programs for a wide range of notions of computation .

Notions of computation and monads

Conditional rewriting logic as a unified model of concurrency

Preliminaries. Categories. Functors. Diagrams. Naturality and Sketches. Products and Sums. Catesian Closed Categories. Finite Discrete Sketches. Limits and Colimits. More About Sketches. Fibrations. Adjoints. Algebras for Endofunctors. Toposes.

/pdf/category-theory-for-computing-science-35hwbxuxwd.pdf

Category Theory For Computing Science

Introduction Prologue: the category of ${\mathbb L}$-spectra Structured ring and module spectra The homotopy theory of $R$-modules The algebraic theory of $R$-modules $R$-ring spectra and the specialization to $MU$ Algebraic $K$-theory of $S$-algebras $R$-algebras and topological model categories Bousfield localizations of $R$-modules and algebras Topological Hochschild homology and cohomology Some basic constructions on spectra Spaces of linear isometries and technical theorems The monadic bar construction Epilogue: The category of ${\mathbb L}$-spectra under $S$ Appendix A. Twisted half-smash products and function spectra Bibliography Index.

Rings, Modules, and Algebras in Stable Homotopy Theory

1. Categories.- 2. Toposes.- 3. Triples.- 4. Theories.- 5. Properties of Toposes.- 6. Permanence Properties of Toposes.- 7. Representation Theorems.- 8. Cocone Theories.- 9. More on Triples.- Index to Exercises.

/pdf/toposes-triples-and-theories-3t0806rx3i.pdf

Toposes, Triples and Theories

Category theory for computing science, 2nd ed.

If 1 G I> E X-4 I -> 1 is a group extension, with t an inclusion, any automorphism T of E which takes G onto itself induces automorphisms T on G and a on 11. However, for a pair (a, T) of automorphism of 11 and G, there may not be an automorphism of E inducing the pair. Let Xx: H -IOut G be the homomorphism induced by the given extension. A pair (a, T) E Aut H1 x Aut G is called compatible if a fixes ker a, and the automorphism induced by a on WIIo is the same as that induced by the inner automorphism of Out G determined by T-. Let C C -H2(H, ZG). The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless "exact" at C in the obvious sense. 1. Notation. If G is a group with subgroup H, we write H< G; if H is normal in G, H<IG. CGH and NGH are the centralizer and normalizer of H in G. Aut G, Inn G, Out G, and ZG are the automorphism group, the inner automorphism group, the outer automorphism group, and the center of G, respectively. If T E Aut G, X denotes its natural image in Out G. If g E G, Ig denotes the inner automorphism of G induced by g: for h E G, hIg = hg =glhg. Maps will be written on the right; composition reads from left to right. If S and T are sets, ST denotes the set of maps from T to S. It is sometimes convenient to write functions exponentially; e.g., x 0 for xp. Throughout the paper, we shall consider fixed groups G, E, Hl for which O. Gi-. ES I f 0 is exact; G and E will be written in additive notation, Hl multiplicatively. G will be identified with its image Gr < E. We fix a left transversal T: H -*I E, specifying 1T= 0. Changing the transversal, and omitting the requirement 1 T= 0 has an effect on almost every statement herein, but the effect can always be calculated explicitly and is ignored. (See Hall [9], Mac Lane [14], or Redei [17].) T can be chosen so that it is a homomorphism if and only if E splits over G. Every element of E can be written uniquely as xT+ a for some x E HI, a E G. (When one writes functions on the left, the form a'+ xT is better.) For x E H, Presented to the Society, January 21, 1971; received by the editors April 21, 1970. AMS 1969 subject classifications. Primary 2022, 2048; Secondary 2050.

/pdf/automorphisms-of-group-extensions-12hvriwhdk.pdf

Automorphisms of group extensions

https://www.jstor.org/stable/pdfplus/2318641.pdf

Charles Wells

Papers

Category Theory For Computing Science

Toposes, Triples and Theories

Category theory for computing science, 2nd ed.

Automorphisms of group extensions

Some applications of the wreath product construction