https://um-catlab.github.io/index/ index /index/ UM CatLab https://um-catlab.github.io/mission/ mission /mission/ Mission Research laboratory at the University of Michigan studying programming languages and semantics. We leverage logic, type theory, and category theory to aid in the design, analysis, and implementation of provably correct software. https://um-catlab.github.io/members/ members /members/ Members https://um-catlab.github.io/ericbond/ ericbond /ericbond/ Eric Bond person https://externalhom.com/ PhD Student 0009-0008-1575-9606 University of Michigan https://um-catlab.github.io/ericgiovannini/ ericgiovannini /ericgiovannini/ Eric Giovannini person https://websites.umich.edu/~ericgio/index.html PhD Student 0009-0003-6871-1714 University of Michigan https://um-catlab.github.io/jesseslater/ jesseslater /jesseslater/ Jesse Slater person PhD Student University of Michigan Coadvised by Xinyu Wang. https://um-catlab.github.io/maxsnew/ maxsnew /maxsnew/ Max S. New person https://maxsnew.com/ Assistant Professor 0000-0001-8141-195X University of Michigan Posts Max S. New 2026 6 2 https://um-catlab.github.io/1WP4/ 1WP4 /1WP4/ Relative Monad definition A monad relative to a profunctor is a function with operations and satisfying three laws Max S. New 2026 6 1 https://um-catlab.github.io/DWP5/ DWP5 /DWP5/ Example of a monad with no strength: Allocation on Indexed Sets theorem Consider the following monad on : Our intuition is that is a simple (but deficient) model of allocation: the represents the number of resources that have been allocated, so that the meaning of types can be dependent on this. Then this monad is an attempt at an allocation monad: an element tells you how many new cells are allocated and gives you a value valid in that future. But this monad does not work for typical denotational semantics because the monad cannot be given a strength with respect to the cartesian product monoidal structure. Aside: The easiest way to verify that this is a monad is to see that the construction is left adjoint to the forgetful and this is the induced monad. In fact this adjunction is monadic. A strength would be a natural . Intuitively such a morphism would send to but the problem is that no such is guaranteed to exist, and therefore a strength cannot be constructed. As a concrete counterexample, pick and where is the Yoneda embedding/Dirac . Note that and . But the strength gives us a morphism . Since is cartesian closed, is strict initial so this implies that but this is false as there is an element . What is the "algebraic" structure that this monad captures? Well as above it is equivalent to a presheaf structure . This can be equivalently stated as a family of maps for every . This can be expressed as a morphism in . So this also provides an example of an algebra with respect to "linear" arities that is not a strong monad. This is why Staton reformulates the notion of a block-algebra, which Plotkin and Power had stated as a similar linear operation. Max S. New 2026 6 1 https://um-catlab.github.io/KAYO/ KAYO /KAYO/ Strong Monad A (left) strength of a monad with respect to a monoidal structure is a natural transformation satisfying certain laws. A concise formulation is that a left-strength is a lifting of from a monad on categories to a monad on actegories, with the monoidal category acting on itself. If is a monoidal closed category, then strength is equivalently an enrichment of . The most general version of this is given by Slattery: a strong monad on multicategories is given by a unit map and an extension map satisfying some laws. This makes the connection to the semantics of effectful programming languages apparent: a monad only has an extension map for unary morphisms but a strength allows for an extension of any single input of a multimorphism. In programming languages terms, the strength is what allows for the use of a monadic bind in the presence of other variables. Despite the common term strong monad, a strength is in general a structure, though in some special cases it is a property. All monads on the category of sets are canonically Set-enriched and therefore carry a strength. Non-trivial strengths abound in (pre)sheaf semantics. Max S. New 2026 5 27 https://um-catlab.github.io/XO0Z/ XO0Z /XO0Z/ Topos of Step-indexed Sets The topos of step-indexed sets, more commonly called the topos of trees is a simple model of step-indexing. There are many equivalent formulations of this category, that all give slightly different definitions of what a step-indexed set is. Presheaves on the ordinal category . Sheaves on with the sheaf condition being that is a one-element set. Sheaves on with the sheaf condition being that the projections from to for all finite is a limit cone. Sheaves on with the sheaf condition being that the projections from to for all finite is a limit cone and that is a one-element set (Assuming Markov's Principle) Sheaves on the topological space whose points are natural numbers and opens are downward-closed subsets. The last two are essentially the same if we identify the downward closed subsets of with their cardinality. The first two are equivalent by identifying the presheaf with the sheaf's . For the remainder of this note we will use definition (4). The global sections functor is a forgetful functor projecting out the set . Under this point of view we can think of the topos of step-indexed sets as a category of sets with structure (though note that the functor is not faithful). The left adjoint is defined as and otherwise. This category is useful in denotational semantics as it supports the definition of guarded recursive functions and types. First we define an endofunctor, , pronounced later or guard. The functor is defined as the restriction of sheaves along the truncated predecessor function , which can be verified to preserve the sheaf condition. Since we get a natural transformation , on components this is just the restriction map . Since is a fixed point of the predecessor function, the global sections functor "erases" the later . A later algebra is defined as a sheaf with a morphism . Expanding the definition this gives us a natural family of functions . We can compose to get an endofunction. The guarded recursion principle says that this endofunction has a unique fixed point satisfying . Under the global sections functor, a later algebra just becomes an endofunction and the guarded fixed point is just a fixed point. In this way, the topos of step-indexed sets can be used as a technical device for recursive definitions in . The free later algebra is the step-indexed analog of the Delay Monad. The global sections functor sends the step-indexed delay monad to the set-based delay monad . This is a resolution of the "algebraicity" of the Delay monad: in Set, the Delay monad is algebraic if and only if Markov's principle holds, but constructively the step-indexed delay monad is "algebraic" but in a generalized sense that it is freely generated by an algebraic operation of non-standard arity. Max S. New 2026 5 26 https://um-catlab.github.io/VWNF/ VWNF /VWNF/ Extended Natural Numbers The extended natural numbers are the final coalgebra for the functor in the category of sets. Concretely these can be constructed as monotone functions from to the booleans. Since the natural numbers are the initial algebra for the same functor we get an canonical injective function . We say an extended natural number is finite if it is in the image of this function. In classical mathematics, the extended natural numbers are simply a coproduct of the natural numbers with a single "infinite" point. Constructively, we can define a function and it being an isomorphism is one of the formulations of Markov's principle. Max S. New 2026 5 26 https://um-catlab.github.io/QCIE/ QCIE /QCIE/ Markov's Principle Markov's Principle is a weak choice principle, weaker than the law of excluded middle. It can be formulated in many equivalent ways: Any sequence of booleans that is not always 0 must contain a 1. The extended natural numbers are equivalent to the natural numbers with a point adjoined. More precisely, the inclusion is surjective (and thus a bijection). Max S. New 2026 5 23 https://um-catlab.github.io/O9R3/ O9R3 /O9R3/ Delay Monad on Set The Delay monad on Sets is given by the final coalgebra . This is monad used as a simple model of programs that may diverge where the number of "steps" is observable. An explicit construction of the Delay monad can be given by in other words, monotone functions from the natural numbers to the maybe monad on where the maybe monad is given the ordering that adjoins a bottom element to a discrete order. The delay monad can be given an inductive characterization as the free fixed point algebra if and only if Markov's principle holds. Without Markov's principle, can be given the structure of a fixed point algebra but we cannot prove it is free. In fact, the delay monad is finitary (and thus the free algebras for an algebraic theory) if and only if Markov's principle holds. Therefore if the delay monad is finitary then it is the free fixed point algebra. Max S. New 2026 5 22 https://um-catlab.github.io/MYZG/ MYZG /MYZG/ Algebraic Theory of an endofunction with a fixed point The algebraic theory of an endofunction with a fixed point consists of two operations satisfying one equation . A computational intuition is that is an internal computation step and that is a diverging computation. The free algebra on can be explicitly described as having underlying set , i.e. an intensional computation either terminates with a result after some finite number of steps, or it diverges. Max S. New 2026 5 21 https://um-catlab.github.io/E1XK/ E1XK /E1XK/ Algebraic Theory of Global Mutable Finite State The algebraic theory of global mutable finite state for a finite set has operations , short for "lookup" For each , short for "update" subject to the equations Intuitively the lookup operation branches on the value of the state, picking the appopriate argument as continuation and update sets the value of the state and then proceeds with the continuation. The equations say that (1) updating the state to what you just read does nothing (2) updating overwrites a preceding update and (3) lookup executes the continuation corresponding to the most recent update. The free state algebra on a set can be constructed explicitly as the stateful functions and the state monad is the corresponding free algebra monad. Max S. New 2026 5 21 https://um-catlab.github.io/L94N/ L94N /L94N/ Finitary Algebraic Theory A finitary signature is a collection of operation symbols with assigned arities which are natural numbers Given a signature, we can form the type of trees . A finitary algebraic theory is a signature with a collection of axioms, which are pairs of trees for finite . An algebra of a theory consists of a carrier , for each operation an operation such that the induced equations from are all satisfied. A homomorphism of algebras is a function that commutes with all of the operations. These form into a category with a forgetful functor A free algebra for a set is a left adjoint to at . Free algebras always exist: they can be constructed as a quotient of , but it is useful when possible to come up with an equivalent description of free algebras as "normal forms" of these quotiented trees. https://um-catlab.github.io/stevenschaefer/ stevenschaefer /stevenschaefer/ Steven Schaefer person https://www.stevenschaefer.net/ University of Michigan 0009-0007-1258-9501 PhD Student Please reach me at stschaef@umich.edu. https://um-catlab.github.io/yuchenjiang/ yuchenjiang /yuchenjiang/ Yuchen Jiang person https://lighghteeloo.github.io/ PhD Student University of Michigan References Context Backlinks Related Contributions