Definition df-termc 50550

Description: Definition of the proper class (termcnex 50653) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 50563). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50592). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50597). See also dftermc3 50608 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50608, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 50568.

Followed directly from the definition, terminal categories are thin (termcthin 50554). The opposite category of a terminal category is "almost" itself (oppctermco 50582). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 50617).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50578) and terminal objects (dftermo4 50579). The universal properties provide an alternate proof of initoeu1 18179, termoeu1 18186, initoeu2 18184, and termoeu2 50315. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50593).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17855, 0thincg 50535, func0g 50166, 0funcg 50162, and initc 50168.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 50549	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1569	. . . . . 6 class 𝑐
4		cbs 17380	. . . . . 6 class Base
5	3, 4	cfv 6537	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1569	. . . . . 6 class 𝑥
8	7	csn 4584	. . . . 5 class {𝑥}
9	5, 8	wceq 1570	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1812	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 50494	. . 3 class ThinCat
12	10, 2, 11	crab 3413	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1570	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is used by:  istermc  50551